EC313 Business and Financial Forecasting Lecture 2

J James Reade

14/01/2015

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\[ \newcommand{\eps}{\varepsilon} \newcommand{\N}[1]{\mathsf{N}\left(#1\right)} \newcommand{\E}[1]{\mathsf{E}\left(#1\right)} \newcommand{\V}[1]{\mathsf{Var}\left(#1\right)} \newcommand{\C}[1]{\mathsf{Cov}\left(#1\right)} \newcommand{\ols}[1]{\widehat{#1}} \newcommand{\cond}[1]{\left|#1\right.} \newcommand{\abs}[1]{\left|#1\right|} \]

Thanks for feedback yesterday!
- Important for me to know you have divided opinions…
- Fast pace inevitable, part of reason for recording lectures.
Feedback from today: http://goo.gl/forms/k7uHCs9XzZ
- Will email with the link, as yesterday.
Project: Term paper, coursework, article.
- Do start thinking, I’m happy to field questions.
Textbook reading (https://www.otexts.org/fpp):
- Yesterday: Chapters 1–2.
- Today: Chapters 3–5.

Introduction

Forecasting is an essential part of life.

This course equips you with.
- Theory: What makes good forecast model?
- Practice: How to create good forecast model?
- Communicating: How to report forecast model?
Online module description here

New Course Structure (with dates)

As informed by email, significant change to structure of course.
Midterm I: Feb 4 11am (week 4): Theory of forecasting.
- Mock by week 2 (please hold me to this!)
Midterm II: March 4 11am (week 8): Forecasting in R.
- Mock by week 6 (please hold me to this!)
Project: ``Forecast a variable of your choice.’’
- Presentation: March 25 (week 11), on your project idea/progress.
  - Guidelines on good presentational skills,
  - Individual feedback will be provided.
- Coursework: Due April 20, 4pm.

Weekly Forecasts

Each week, submit your forecast online. For week 1, submit here.
- Deadline each week: Thursday 6pm. Late forecasts will be penalised.
Why so many?
- Forecast different time horizons.
- Check on progress through term.
Additional series each week:
- Week 1: Rightmove House Price Index (YoY) (Jan)
- Week 2: GDP 2015 Q4 first estimate (released Jan 27 9:30am)
- Week 3: Mortgage approvals (released by BoE Jan 30)

FTSE

Forecast Friday’s FTSE closing price.
Access data: https://uk.finance.yahoo.com/q/hp?s=%5EFTSE

FTSE — Issues

It’s very persistent (studies have found random walk best).
Random walk forecast means longer horizon forecasts usually worse.
Forecast failure here is no badge of shame.

Google Trends and Dave

Data from http://goo.gl/HeOxdw
- Top right menu, Download as CSV.
- Open in Excel to turn into correct format.

Issues with Dave

Only one previous election to really use as indicative.
Difficult to think about predictive variables.
It’s relative scale — a few 100s mean new heights of popularity.
Forecasts will be assessed on the number for a completed week, the following week.
- Hence not if the relative scales are revised nearer the election.

Rightmove House Price Index (YoY) (Jan)

Data from http://www.fxstreet.com/economic-calendar/

Issues with House Price Index

No consensus.
In fxstreet data, “previous” is naive forecast.
- Close by nature of persistence of data.
- But always too high when falling, too low when rising.

Course Outline

Week 1
- Tuesday: Introduction to forecasting (and to R) (Ch. 1–2).
- Wednesday: Judgemental (Ch. 3) and Regression-based Forecasts (Ch. 4–5).
Week 2
- Tuesday: ARIMA Models (Ch. 8), Non-stationarity and structural breaks.
- Wednesday: Model selection and impulse saturation.
Week 3
- Tuesday: Structural Time Series Modelling (Ch. 6) and Exponential Smoothing Methods (Ch. 7).
- Wednesday: Course recap.
Week 4, Tuesday: Midterm I

Week 4
- Wednesday: Practical Introduction to R.
Week 5
- Tuesday: Introduction to R: Running regression models.
- Wednesday: R Class I: BYOD (bring your own data).
Week 6: Enhancement Week (forecast competition and Settlers)
Week 7
- Tuesday: Introduction to R: Forecasting.
- Wednesday: R Class II: BYOD (bring your own data).
Week 8
- Tuesday: Introduction to R: Evaluating forecasts.
- Wednesday: Midterm II

Week 9
- Tuesday: Project: How to conduct an empirical project.
- Wednesday: Project Class I: Drop-in sessions.
Week 10
- Tuesday: Project: Example ideas I.
- Wednesday: Project Class II: Drop-in sessions.
Week 11
- Tuesday: Project: Example ideas II and Presentation Prep.
- Wednesday: Project Presentations

Yesterday: Introduction to forecasting

Data
- What can we forecast?
- What do we do with data?
- Examples
Forecasting
- What is a forecast?
- Forecast errors
- Optimal forecasts
- Forecast failure
- Basic Forecast models

Today: Judgemental and Regression-based Forecasts

Judgemental Forecasting (Ch. 3):
- Limitations
- Key principles
- Delphi method
- Scenario forecasting
Regression-based forecasts (Chs. 4–5)
- OLS
- Regression and correlation
- Evaluating the model/residual diagnostics
- Useful predictors
- Inference

Judgemental Forecasting (Ch. 3)

Sometimes data simply does not exist: No historical precedent.
- E.g. iPad, new immigration restrictions.
Also: Data may not be available yet: e.g. quarterly GDP.
Sometimes data available yet adjustments still made:
- Principle: If facts inconvenient, dispose of them…
- Experts may add finishing touch to work of analysts.
- Econometricians consider many different models before choosing ‘best’.
Judgemental forecasting best when Lawrence et at (2006):¹
- Adjuster has important (“expert”) knowledge.
- Adjuster has timely up-to-date information.

(Sec. 3.2) Beware of Limitations

Judgemental forecasts subjective, unlikely unbiased.
Also inconsistent:
- Statistical methods formula based, converge on true values.
- Human methods rely on cognition: Limited memory.
Human methods also affected by various biases we exhibit:
- Political or personal agendas.
- Optimism.
- Anchoring.

(Sec. 3.3) Key Principles

Important for own judgemental forecasting approach.
Set task clearly and consicely.
- E.g. avoid emotive terms (“benefits tourists”).
Implement systematic approach:
- Set out systematically information to be used.
Document and Justify:
- Write out decision rules, assumptions made.
Systematically evaluate all forecasts made.
Ensure forecasts and targets not confused.

(Sec. 3.4) The Delphi Method

Developed in 1950s at RAND Corporation (Linstone and Turoff, 1975):²
- Group forecasts generally more accurate than those from individuals.
- Construct (expert) consensus group forecasts by iterative process.
Involves following stages:
- Construct panel of (anonymous) experts.
- Forecasting task(s) set to panel.
- Forecasts compiled, summarised and feedback distributed.
- Forecasters can review and resubmit their forecasts.
- Final forecasts constructed from panel forecasts.

(Sec. 3.6) Scenario Forecasting

Scenario forecasting somewhat detached from standard appraisal methods:
- Worst-case scenario may have very low likelihood of occurring.
Scenario forecasting common with policy analysis, product development.
- Climate change forecasting also.
Analogous to forecasting with explanatory variables:
- Need some scenario for all explanatory variables into future.

Regression-based Forecasting

Assuming data exists, and can be utilised, we can and should use it.
Optimal method is some formal statistical modelling:
- Less (though not totally un-)objective.
Forecast \(y\) assuming relationship between \(y\), \(x\).
Simple = one regressor. Later, more than one.
Jargon:
- \(y\) is forecast/dependent/explained variable or regressand.
- \(x\) is predictor/independent/explanatory variable or regressor or covariate.

Simple linear model

Assume bivariate linear model: \[ y = \beta_0 + \beta_1 x + \varepsilon. \]

Simple linear model

4.1 Simple Linear Model

Residuals from model are \(\ols{\eps}_t=y_t-\ols{y}_t\).
One-step forecast errors: \(\ols{\eps}_{T+1\cond{T}}=y_{T+1}-\ols{y}_{T+1\cond{T}}\).
Residuals same as one-step forecast errors: Test vs training.
Residual assumptions:
- \(\E{\ols{\eps}_{T+h\cond{T}}}=0\),
- \(\V{\ols{\eps}_{T+h\cond{T}}}=\sigma^2\),
- \(\C{\ols{\eps}_{T+h\cond{T}},\ols{e}_{T+j\cond{T}}}=0\), \(h\neq j\).
If these fails, should try to improve model.
- Model satisfying these assumption could be improved.
- Calculation of forecast error bounds invalidated by failure of these assumptions.

4.2 Least Squares Estimation

Want to estimate \(\beta_0\) and \(\beta_1\) in order to forecast.
Least squares principle minimises squared errors: \[ \sum_{i=1}^N \varepsilon_t^2 = \sum_{i=1}^N (y_t - \beta_0 - \beta_1x_t)^2. \]
Yields estimators: \[ \widehat{\beta}_1=\frac{ \sum_{i=1}^{N}(y_t-\bar{y})(x_t-\bar{x})}{\sum_{i=1}^{N}(x_t-\bar{x})^2},\\ \widehat{\beta}_0=\bar{y}-\widehat{\beta}_1\bar{x}. \]

4.2 OLS: Truth and Statistics

We work with samples and cannot know the truth.
In the picture below, the truth is known: The statistical distribution the data are drawn from.
But with only 100 replications, the OLS line (green) is not the same as the actual line (red).
- We do not have the population.
- Sample variation exists.

4.2 Least Squares Estimation

Fitted values: \[ \ols{y}_t = \ols{\beta}_0 + \ols{\beta}_1 x_t. \]
Residuals: \[ e_t = y_t - \widehat{y}_t = y_t -\widehat{\beta}_0-\widehat{\beta}_1x_t. \]

4.3 Regression and Correlation

Slope coefficient \(\beta_1\) and correlation coefficient closely related: \[ \hat{\beta}_1=r\frac{s_{y}}{s_x}. \]
- Regression model asserts causal relationship, unlike correlation.

4.4 Evaluating Regression Models

Plotting residuals: Aim is \(\ols{\eps}_t\) has same properties as \(\eps_t\).
- Systematic patterns alert us to problems.
Outliers: Observations that take on extreme values relative to rest of sample.
- Can distort inference by influencing slope and intercept parameters.
- Caution: Seeming outliers (e.g. Xmas Day) may not generate large residuals.

4.4 Evaluating Regression Models

Goodness-of-fit measures: \[ R^2 = \frac{\sum(\hat{y}_{t} - \bar{y})^2}{\sum(y_{t}-\bar{y})^2}, \]
Standard error of regression: \[ s_e=\sqrt{\frac{1}{N-2}\sum_{i=1}^{N}{e_t^2}}. \]

R^2 and standard errors

Below: green lines are residuals, blue lines differences from mean.
- Model leaves green bits unexplained, mean leaves blue bits unexplained.
- If model better than mean, green lines smaller, and \(R^2\)>0.

4.4 Evaluating Regression Models: Outliers

Outliers: Observations that do not belong to the same statistical distribution as the rest of the data.
Outliers distort inference: plot below shows the OLS line is affected.

Outliers, the Constant Term, and Forecasting

Outliers affect the constant term.
- Constant term impacts on forecasts (it is essentially the mean).
- Thus detection of outliers is important.
Random walk forecast uses final observation to forecast:
- What if final observation is an outlier?
Next Tuesday we develop methods to cope with these problems.

4.5 Forecasting with Regression

Compute forecasts as we compute fitted values (on training set): \[ \ols{y} = \ols{\beta}_0 + \ols{\beta}_1 x \]
- If we use \(x\) from test set this is forecast.

4.7 Non-Linear Functional Forms

Often need non-linear transformations to forecast better.
Most common transformation: Log — but which variable? E.g.: \[ \log y_t=\beta_0+\beta_1 \log x_t + \varepsilon_t. \]

Model	Dependent Variable	Independent Variable	Interpretation of \(\beta_1\)
Level-level	\(y\)	\(x\)	\(\Delta y = \beta_1 \Delta x\)
Level-log	\(y\)	\(\log(x)\)	\(\Delta y = (\beta_1/100) \% \Delta x\)
Log-level	\(\log(y)\)	\(x\)	\(\% \Delta y = 100\beta_1 \Delta x\)
Log-log	\(\log(y)\)	\(\log(x)\)	\(\% \Delta y = \beta_1 \% \Delta x\)

4.8 Regression with Time Series Data

While our examples have been time series thus far, we must be careful.
Example: Forecasting US consumption using disposable income levels: \[ cons_t = \beta_0 + \beta_1 inc_t + u_t, \]
Problem: To forecast, say, 2014Q4 (in the past!), need income in 2014Q4.
- Scenario forecasting: Use scenarios for \(inc_t\) to forecasting \(cons_t\).
- Ex ante forecast: Uses forecast values for \(inc_t\).
- Ex post forecast: Uses actual values for \(inc_t\).

4.8 Forecasting (with scenarios)

We need to input some values for our \(x\) variables:
- One solution is to input scenarios.
Below: Range of forecasts for unemployment rate with vacancy rate varying from 0 to 5%.

4.8 Ex-Post Forecasting

Ex-post forecasting is holding back observation to forecast.

##                  date unrate
## 2000-12-01 2000-12-01    3.9
## 2001-01-01 2001-01-01    4.2
## 2001-02-01 2001-02-01    4.2
## 2001-03-01 2001-03-01    4.3
## 2001-04-01 2001-04-01    4.4
## 2001-05-01 2001-05-01    4.3
## 2001-06-01 2001-06-01    4.5
## 2001-07-01 2001-07-01    4.6
## 2001-08-01 2001-08-01    4.9
## 2001-09-01 2001-09-01    5.0
## 2001-10-01 2001-10-01    5.3
## 2001-11-01 2001-11-01    5.5
## 2001-12-01 2001-12-01    5.7
## 2002-01-01 2002-01-01    5.7
## 2002-02-01 2002-02-01    5.7
## 2002-03-01 2002-03-01    5.7
## 2002-04-01 2002-04-01    5.9
## 2002-05-01 2002-05-01    5.8
## 2002-06-01 2002-06-01    5.8
## 2002-07-01 2002-07-01    5.8
## 2002-08-01 2002-08-01    5.7
## 2002-09-01 2002-09-01    5.7
## 2002-10-01 2002-10-01    5.7
## 2002-11-01 2002-11-01    5.9
## 2002-12-01 2002-12-01    6.0
## 2003-01-01 2003-01-01    5.8
## 2003-02-01 2003-02-01    5.9
## 2003-03-01 2003-03-01    5.9
## 2003-04-01 2003-04-01    6.0
## 2003-05-01 2003-05-01    6.1
## 2003-06-01 2003-06-01    6.3
## 2003-07-01 2003-07-01    6.2
## 2003-08-01 2003-08-01    6.1
## 2003-09-01 2003-09-01    6.1
## 2003-10-01 2003-10-01    6.0
## 2003-11-01 2003-11-01    5.8
## 2003-12-01 2003-12-01    5.7
## 2004-01-01 2004-01-01    5.7
## 2004-02-01 2004-02-01    5.6
## 2004-03-01 2004-03-01    5.8
## 2004-04-01 2004-04-01    5.6
## 2004-05-01 2004-05-01    5.6
## 2004-06-01 2004-06-01    5.6
## 2004-07-01 2004-07-01    5.5
## 2004-08-01 2004-08-01    5.4
## 2004-09-01 2004-09-01    5.4
## 2004-10-01 2004-10-01    5.5
## 2004-11-01 2004-11-01    5.4
## 2004-12-01 2004-12-01    5.4
## 2005-01-01 2005-01-01    5.3
## 2005-02-01 2005-02-01    5.4
## 2005-03-01 2005-03-01    5.2
## 2005-04-01 2005-04-01    5.2
## 2005-05-01 2005-05-01    5.1
## 2005-06-01 2005-06-01    5.0
## 2005-07-01 2005-07-01    5.0
## 2005-08-01 2005-08-01    4.9
## 2005-09-01 2005-09-01    5.0
## 2005-10-01 2005-10-01    5.0
## 2005-11-01 2005-11-01    5.0
## 2005-12-01 2005-12-01    4.9
## 2006-01-01 2006-01-01    4.7
## 2006-02-01 2006-02-01    4.8
## 2006-03-01 2006-03-01    4.7
## 2006-04-01 2006-04-01    4.7
## 2006-05-01 2006-05-01    4.6
## 2006-06-01 2006-06-01    4.6
## 2006-07-01 2006-07-01    4.7
## 2006-08-01 2006-08-01    4.7
## 2006-09-01 2006-09-01    4.5
## 2006-10-01 2006-10-01    4.4
## 2006-11-01 2006-11-01    4.5
## 2006-12-01 2006-12-01    4.4
## 2007-01-01 2007-01-01    4.6
## 2007-02-01 2007-02-01    4.5
## 2007-03-01 2007-03-01    4.4
## 2007-04-01 2007-04-01    4.5
## 2007-05-01 2007-05-01    4.4
## 2007-06-01 2007-06-01    4.6
## 2007-07-01 2007-07-01    4.7
## 2007-08-01 2007-08-01    4.6
## 2007-09-01 2007-09-01    4.7
## 2007-10-01 2007-10-01    4.7
## 2007-11-01 2007-11-01    4.7
## 2007-12-01 2007-12-01    5.0
## 2008-01-01 2008-01-01    5.0
## 2008-02-01 2008-02-01    4.9
## 2008-03-01 2008-03-01    5.1
## 2008-04-01 2008-04-01    5.0
## 2008-05-01 2008-05-01    5.4
## 2008-06-01 2008-06-01    5.6
## 2008-07-01 2008-07-01    5.8
## 2008-08-01 2008-08-01    6.1
## 2008-09-01 2008-09-01    6.1
## 2008-10-01 2008-10-01    6.5
## 2008-11-01 2008-11-01    6.8
## 2008-12-01 2008-12-01    7.3
## 2009-01-01 2009-01-01    7.8
## 2009-02-01 2009-02-01    8.3
## 2009-03-01 2009-03-01    8.7
## 2009-04-01 2009-04-01    9.0
## 2009-05-01 2009-05-01    9.4
## 2009-06-01 2009-06-01    9.5
## 2009-07-01 2009-07-01    9.5
## 2009-08-01 2009-08-01    9.6
## 2009-09-01 2009-09-01    9.8
## 2009-10-01 2009-10-01   10.0
## 2009-11-01 2009-11-01    9.9
## 2009-12-01 2009-12-01    9.9
## 2010-01-01 2010-01-01    9.8
## 2010-02-01 2010-02-01    9.8
## 2010-03-01 2010-03-01    9.9
## 2010-04-01 2010-04-01    9.9
## 2010-05-01 2010-05-01    9.6
## 2010-06-01 2010-06-01    9.4
## 2010-07-01 2010-07-01    9.4
## 2010-08-01 2010-08-01    9.5
## 2010-09-01 2010-09-01    9.5
## 2010-10-01 2010-10-01    9.4
## 2010-11-01 2010-11-01    9.8
## 2010-12-01 2010-12-01    9.3
## 2011-01-01 2011-01-01    9.2
## 2011-02-01 2011-02-01    9.0
## 2011-03-01 2011-03-01    9.0
## 2011-04-01 2011-04-01    9.1
## 2011-05-01 2011-05-01    9.0
## 2011-06-01 2011-06-01    9.1
## 2011-07-01 2011-07-01    9.0
## 2011-08-01 2011-08-01    9.0
## 2011-09-01 2011-09-01    9.0
## 2011-10-01 2011-10-01    8.8
## 2011-11-01 2011-11-01    8.6
## 2011-12-01 2011-12-01    8.5
## 2012-01-01 2012-01-01    8.3
## 2012-02-01 2012-02-01    8.3
## 2012-03-01 2012-03-01    8.2
## 2012-04-01 2012-04-01    8.2
## 2012-05-01 2012-05-01    8.2
## 2012-06-01 2012-06-01    8.2
## 2012-07-01 2012-07-01    8.2
## 2012-08-01 2012-08-01    8.0
## 2012-09-01 2012-09-01    7.8
## 2012-10-01 2012-10-01    7.8
## 2012-11-01 2012-11-01    7.7
## 2012-12-01 2012-12-01    7.9
## 2013-01-01 2013-01-01    8.0
## 2013-02-01 2013-02-01    7.7
## 2013-03-01 2013-03-01    7.5
## 2013-04-01 2013-04-01    7.6
## 2013-05-01 2013-05-01    7.5
## 2013-06-01 2013-06-01    7.5
## 2013-07-01 2013-07-01    7.3
## 2013-08-01 2013-08-01    7.2
## 2013-09-01 2013-09-01    7.2
## 2013-10-01 2013-10-01    7.2
## 2013-11-01 2013-11-01    7.0
## 2013-12-01 2013-12-01    6.7
## 2014-01-01 2014-01-01    6.6
## 2014-02-01 2014-02-01    6.7
## 2014-03-01 2014-03-01    6.6
## 2014-04-01 2014-04-01    6.2
## 2014-05-01 2014-05-01    6.3
## 2014-06-01 2014-06-01    6.1
## 2014-07-01 2014-07-01    6.2
## 2014-08-01 2014-08-01    6.1
## 2014-09-01 2014-09-01    5.9
## 2014-10-01 2014-10-01    5.7
## 2014-11-01 2014-11-01    5.8
## 2014-12-01 2014-12-01    5.6
## 2015-01-01 2015-01-01    5.7
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
##                  date unrate vacrate
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
## 
## Call:
## lm(formula = unrate ~ vacrate, data = data.0)
## 
## Coefficients:
## (Intercept)      vacrate  
##      15.042       -3.173  
## 
##   Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1       4.253769 2.786695 5.720842 2.002591 6.504947
## 2       4.253769 2.786695 5.720842 2.002591 6.504947

##                  date unrate
## 2000-12-01 2000-12-01    3.9
## 2001-01-01 2001-01-01    4.2
## 2001-02-01 2001-02-01    4.2
## 2001-03-01 2001-03-01    4.3
## 2001-04-01 2001-04-01    4.4
## 2001-05-01 2001-05-01    4.3
## 2001-06-01 2001-06-01    4.5
## 2001-07-01 2001-07-01    4.6
## 2001-08-01 2001-08-01    4.9
## 2001-09-01 2001-09-01    5.0
## 2001-10-01 2001-10-01    5.3
## 2001-11-01 2001-11-01    5.5
## 2001-12-01 2001-12-01    5.7
## 2002-01-01 2002-01-01    5.7
## 2002-02-01 2002-02-01    5.7
## 2002-03-01 2002-03-01    5.7
## 2002-04-01 2002-04-01    5.9
## 2002-05-01 2002-05-01    5.8
## 2002-06-01 2002-06-01    5.8
## 2002-07-01 2002-07-01    5.8
## 2002-08-01 2002-08-01    5.7
## 2002-09-01 2002-09-01    5.7
## 2002-10-01 2002-10-01    5.7
## 2002-11-01 2002-11-01    5.9
## 2002-12-01 2002-12-01    6.0
## 2003-01-01 2003-01-01    5.8
## 2003-02-01 2003-02-01    5.9
## 2003-03-01 2003-03-01    5.9
## 2003-04-01 2003-04-01    6.0
## 2003-05-01 2003-05-01    6.1
## 2003-06-01 2003-06-01    6.3
## 2003-07-01 2003-07-01    6.2
## 2003-08-01 2003-08-01    6.1
## 2003-09-01 2003-09-01    6.1
## 2003-10-01 2003-10-01    6.0
## 2003-11-01 2003-11-01    5.8
## 2003-12-01 2003-12-01    5.7
## 2004-01-01 2004-01-01    5.7
## 2004-02-01 2004-02-01    5.6
## 2004-03-01 2004-03-01    5.8
## 2004-04-01 2004-04-01    5.6
## 2004-05-01 2004-05-01    5.6
## 2004-06-01 2004-06-01    5.6
## 2004-07-01 2004-07-01    5.5
## 2004-08-01 2004-08-01    5.4
## 2004-09-01 2004-09-01    5.4
## 2004-10-01 2004-10-01    5.5
## 2004-11-01 2004-11-01    5.4
## 2004-12-01 2004-12-01    5.4
## 2005-01-01 2005-01-01    5.3
## 2005-02-01 2005-02-01    5.4
## 2005-03-01 2005-03-01    5.2
## 2005-04-01 2005-04-01    5.2
## 2005-05-01 2005-05-01    5.1
## 2005-06-01 2005-06-01    5.0
## 2005-07-01 2005-07-01    5.0
## 2005-08-01 2005-08-01    4.9
## 2005-09-01 2005-09-01    5.0
## 2005-10-01 2005-10-01    5.0
## 2005-11-01 2005-11-01    5.0
## 2005-12-01 2005-12-01    4.9
## 2006-01-01 2006-01-01    4.7
## 2006-02-01 2006-02-01    4.8
## 2006-03-01 2006-03-01    4.7
## 2006-04-01 2006-04-01    4.7
## 2006-05-01 2006-05-01    4.6
## 2006-06-01 2006-06-01    4.6
## 2006-07-01 2006-07-01    4.7
## 2006-08-01 2006-08-01    4.7
## 2006-09-01 2006-09-01    4.5
## 2006-10-01 2006-10-01    4.4
## 2006-11-01 2006-11-01    4.5
## 2006-12-01 2006-12-01    4.4
## 2007-01-01 2007-01-01    4.6
## 2007-02-01 2007-02-01    4.5
## 2007-03-01 2007-03-01    4.4
## 2007-04-01 2007-04-01    4.5
## 2007-05-01 2007-05-01    4.4
## 2007-06-01 2007-06-01    4.6
## 2007-07-01 2007-07-01    4.7
## 2007-08-01 2007-08-01    4.6
## 2007-09-01 2007-09-01    4.7
## 2007-10-01 2007-10-01    4.7
## 2007-11-01 2007-11-01    4.7
## 2007-12-01 2007-12-01    5.0
## 2008-01-01 2008-01-01    5.0
## 2008-02-01 2008-02-01    4.9
## 2008-03-01 2008-03-01    5.1
## 2008-04-01 2008-04-01    5.0
## 2008-05-01 2008-05-01    5.4
## 2008-06-01 2008-06-01    5.6
## 2008-07-01 2008-07-01    5.8
## 2008-08-01 2008-08-01    6.1
## 2008-09-01 2008-09-01    6.1
## 2008-10-01 2008-10-01    6.5
## 2008-11-01 2008-11-01    6.8
## 2008-12-01 2008-12-01    7.3
## 2009-01-01 2009-01-01    7.8
## 2009-02-01 2009-02-01    8.3
## 2009-03-01 2009-03-01    8.7
## 2009-04-01 2009-04-01    9.0
## 2009-05-01 2009-05-01    9.4
## 2009-06-01 2009-06-01    9.5
## 2009-07-01 2009-07-01    9.5
## 2009-08-01 2009-08-01    9.6
## 2009-09-01 2009-09-01    9.8
## 2009-10-01 2009-10-01   10.0
## 2009-11-01 2009-11-01    9.9
## 2009-12-01 2009-12-01    9.9
## 2010-01-01 2010-01-01    9.8
## 2010-02-01 2010-02-01    9.8
## 2010-03-01 2010-03-01    9.9
## 2010-04-01 2010-04-01    9.9
## 2010-05-01 2010-05-01    9.6
## 2010-06-01 2010-06-01    9.4
## 2010-07-01 2010-07-01    9.4
## 2010-08-01 2010-08-01    9.5
## 2010-09-01 2010-09-01    9.5
## 2010-10-01 2010-10-01    9.4
## 2010-11-01 2010-11-01    9.8
## 2010-12-01 2010-12-01    9.3
## 2011-01-01 2011-01-01    9.2
## 2011-02-01 2011-02-01    9.0
## 2011-03-01 2011-03-01    9.0
## 2011-04-01 2011-04-01    9.1
## 2011-05-01 2011-05-01    9.0
## 2011-06-01 2011-06-01    9.1
## 2011-07-01 2011-07-01    9.0
## 2011-08-01 2011-08-01    9.0
## 2011-09-01 2011-09-01    9.0
## 2011-10-01 2011-10-01    8.8
## 2011-11-01 2011-11-01    8.6
## 2011-12-01 2011-12-01    8.5
## 2012-01-01 2012-01-01    8.3
## 2012-02-01 2012-02-01    8.3
## 2012-03-01 2012-03-01    8.2
## 2012-04-01 2012-04-01    8.2
## 2012-05-01 2012-05-01    8.2
## 2012-06-01 2012-06-01    8.2
## 2012-07-01 2012-07-01    8.2
## 2012-08-01 2012-08-01    8.0
## 2012-09-01 2012-09-01    7.8
## 2012-10-01 2012-10-01    7.8
## 2012-11-01 2012-11-01    7.7
## 2012-12-01 2012-12-01    7.9
## 2013-01-01 2013-01-01    8.0
## 2013-02-01 2013-02-01    7.7
## 2013-03-01 2013-03-01    7.5
## 2013-04-01 2013-04-01    7.6
## 2013-05-01 2013-05-01    7.5
## 2013-06-01 2013-06-01    7.5
## 2013-07-01 2013-07-01    7.3
## 2013-08-01 2013-08-01    7.2
## 2013-09-01 2013-09-01    7.2
## 2013-10-01 2013-10-01    7.2
## 2013-11-01 2013-11-01    7.0
## 2013-12-01 2013-12-01    6.7
## 2014-01-01 2014-01-01    6.6
## 2014-02-01 2014-02-01    6.7
## 2014-03-01 2014-03-01    6.6
## 2014-04-01 2014-04-01    6.2
## 2014-05-01 2014-05-01    6.3
## 2014-06-01 2014-06-01    6.1
## 2014-07-01 2014-07-01    6.2
## 2014-08-01 2014-08-01    6.1
## 2014-09-01 2014-09-01    5.9
## 2014-10-01 2014-10-01    5.7
## 2014-11-01 2014-11-01    5.8
## 2014-12-01 2014-12-01    5.6
## 2015-01-01 2015-01-01    5.7
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
##                  date unrate vacrate
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
## 
## Call:
## lm(formula = unrate ~ vacrate, data = data.0)
## 
## Coefficients:
## (Intercept)      vacrate  
##      15.572       -3.393  
## 
##   Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1       4.374072 2.937740 5.810403 2.169790 6.578354
## 2       4.374072 2.937740 5.810403 2.169790 6.578354
## 3       4.034744 2.594939 5.474549 1.825132 6.244357
## 4       4.713399 3.280046 6.146752 2.513688 6.913110
## 5       4.034744 2.594939 5.474549 1.825132 6.244357
## 6       4.034744 2.594939 5.474549 1.825132 6.244357
## 7       4.034744 2.594939 5.474549 1.825132 6.244357
## 8       4.034744 2.594939 5.474549 1.825132 6.244357

##                  date unrate
## 2000-12-01 2000-12-01    3.9
## 2001-01-01 2001-01-01    4.2
## 2001-02-01 2001-02-01    4.2
## 2001-03-01 2001-03-01    4.3
## 2001-04-01 2001-04-01    4.4
## 2001-05-01 2001-05-01    4.3
## 2001-06-01 2001-06-01    4.5
## 2001-07-01 2001-07-01    4.6
## 2001-08-01 2001-08-01    4.9
## 2001-09-01 2001-09-01    5.0
## 2001-10-01 2001-10-01    5.3
## 2001-11-01 2001-11-01    5.5
## 2001-12-01 2001-12-01    5.7
## 2002-01-01 2002-01-01    5.7
## 2002-02-01 2002-02-01    5.7
## 2002-03-01 2002-03-01    5.7
## 2002-04-01 2002-04-01    5.9
## 2002-05-01 2002-05-01    5.8
## 2002-06-01 2002-06-01    5.8
## 2002-07-01 2002-07-01    5.8
## 2002-08-01 2002-08-01    5.7
## 2002-09-01 2002-09-01    5.7
## 2002-10-01 2002-10-01    5.7
## 2002-11-01 2002-11-01    5.9
## 2002-12-01 2002-12-01    6.0
## 2003-01-01 2003-01-01    5.8
## 2003-02-01 2003-02-01    5.9
## 2003-03-01 2003-03-01    5.9
## 2003-04-01 2003-04-01    6.0
## 2003-05-01 2003-05-01    6.1
## 2003-06-01 2003-06-01    6.3
## 2003-07-01 2003-07-01    6.2
## 2003-08-01 2003-08-01    6.1
## 2003-09-01 2003-09-01    6.1
## 2003-10-01 2003-10-01    6.0
## 2003-11-01 2003-11-01    5.8
## 2003-12-01 2003-12-01    5.7
## 2004-01-01 2004-01-01    5.7
## 2004-02-01 2004-02-01    5.6
## 2004-03-01 2004-03-01    5.8
## 2004-04-01 2004-04-01    5.6
## 2004-05-01 2004-05-01    5.6
## 2004-06-01 2004-06-01    5.6
## 2004-07-01 2004-07-01    5.5
## 2004-08-01 2004-08-01    5.4
## 2004-09-01 2004-09-01    5.4
## 2004-10-01 2004-10-01    5.5
## 2004-11-01 2004-11-01    5.4
## 2004-12-01 2004-12-01    5.4
## 2005-01-01 2005-01-01    5.3
## 2005-02-01 2005-02-01    5.4
## 2005-03-01 2005-03-01    5.2
## 2005-04-01 2005-04-01    5.2
## 2005-05-01 2005-05-01    5.1
## 2005-06-01 2005-06-01    5.0
## 2005-07-01 2005-07-01    5.0
## 2005-08-01 2005-08-01    4.9
## 2005-09-01 2005-09-01    5.0
## 2005-10-01 2005-10-01    5.0
## 2005-11-01 2005-11-01    5.0
## 2005-12-01 2005-12-01    4.9
## 2006-01-01 2006-01-01    4.7
## 2006-02-01 2006-02-01    4.8
## 2006-03-01 2006-03-01    4.7
## 2006-04-01 2006-04-01    4.7
## 2006-05-01 2006-05-01    4.6
## 2006-06-01 2006-06-01    4.6
## 2006-07-01 2006-07-01    4.7
## 2006-08-01 2006-08-01    4.7
## 2006-09-01 2006-09-01    4.5
## 2006-10-01 2006-10-01    4.4
## 2006-11-01 2006-11-01    4.5
## 2006-12-01 2006-12-01    4.4
## 2007-01-01 2007-01-01    4.6
## 2007-02-01 2007-02-01    4.5
## 2007-03-01 2007-03-01    4.4
## 2007-04-01 2007-04-01    4.5
## 2007-05-01 2007-05-01    4.4
## 2007-06-01 2007-06-01    4.6
## 2007-07-01 2007-07-01    4.7
## 2007-08-01 2007-08-01    4.6
## 2007-09-01 2007-09-01    4.7
## 2007-10-01 2007-10-01    4.7
## 2007-11-01 2007-11-01    4.7
## 2007-12-01 2007-12-01    5.0
## 2008-01-01 2008-01-01    5.0
## 2008-02-01 2008-02-01    4.9
## 2008-03-01 2008-03-01    5.1
## 2008-04-01 2008-04-01    5.0
## 2008-05-01 2008-05-01    5.4
## 2008-06-01 2008-06-01    5.6
## 2008-07-01 2008-07-01    5.8
## 2008-08-01 2008-08-01    6.1
## 2008-09-01 2008-09-01    6.1
## 2008-10-01 2008-10-01    6.5
## 2008-11-01 2008-11-01    6.8
## 2008-12-01 2008-12-01    7.3
## 2009-01-01 2009-01-01    7.8
## 2009-02-01 2009-02-01    8.3
## 2009-03-01 2009-03-01    8.7
## 2009-04-01 2009-04-01    9.0
## 2009-05-01 2009-05-01    9.4
## 2009-06-01 2009-06-01    9.5
## 2009-07-01 2009-07-01    9.5
## 2009-08-01 2009-08-01    9.6
## 2009-09-01 2009-09-01    9.8
## 2009-10-01 2009-10-01   10.0
## 2009-11-01 2009-11-01    9.9
## 2009-12-01 2009-12-01    9.9
## 2010-01-01 2010-01-01    9.8
## 2010-02-01 2010-02-01    9.8
## 2010-03-01 2010-03-01    9.9
## 2010-04-01 2010-04-01    9.9
## 2010-05-01 2010-05-01    9.6
## 2010-06-01 2010-06-01    9.4
## 2010-07-01 2010-07-01    9.4
## 2010-08-01 2010-08-01    9.5
## 2010-09-01 2010-09-01    9.5
## 2010-10-01 2010-10-01    9.4
## 2010-11-01 2010-11-01    9.8
## 2010-12-01 2010-12-01    9.3
## 2011-01-01 2011-01-01    9.2
## 2011-02-01 2011-02-01    9.0
## 2011-03-01 2011-03-01    9.0
## 2011-04-01 2011-04-01    9.1
## 2011-05-01 2011-05-01    9.0
## 2011-06-01 2011-06-01    9.1
## 2011-07-01 2011-07-01    9.0
## 2011-08-01 2011-08-01    9.0
## 2011-09-01 2011-09-01    9.0
## 2011-10-01 2011-10-01    8.8
## 2011-11-01 2011-11-01    8.6
## 2011-12-01 2011-12-01    8.5
## 2012-01-01 2012-01-01    8.3
## 2012-02-01 2012-02-01    8.3
## 2012-03-01 2012-03-01    8.2
## 2012-04-01 2012-04-01    8.2
## 2012-05-01 2012-05-01    8.2
## 2012-06-01 2012-06-01    8.2
## 2012-07-01 2012-07-01    8.2
## 2012-08-01 2012-08-01    8.0
## 2012-09-01 2012-09-01    7.8
## 2012-10-01 2012-10-01    7.8
## 2012-11-01 2012-11-01    7.7
## 2012-12-01 2012-12-01    7.9
## 2013-01-01 2013-01-01    8.0
## 2013-02-01 2013-02-01    7.7
## 2013-03-01 2013-03-01    7.5
## 2013-04-01 2013-04-01    7.6
## 2013-05-01 2013-05-01    7.5
## 2013-06-01 2013-06-01    7.5
## 2013-07-01 2013-07-01    7.3
## 2013-08-01 2013-08-01    7.2
## 2013-09-01 2013-09-01    7.2
## 2013-10-01 2013-10-01    7.2
## 2013-11-01 2013-11-01    7.0
## 2013-12-01 2013-12-01    6.7
## 2014-01-01 2014-01-01    6.6
## 2014-02-01 2014-02-01    6.7
## 2014-03-01 2014-03-01    6.6
## 2014-04-01 2014-04-01    6.2
## 2014-05-01 2014-05-01    6.3
## 2014-06-01 2014-06-01    6.1
## 2014-07-01 2014-07-01    6.2
## 2014-08-01 2014-08-01    6.1
## 2014-09-01 2014-09-01    5.9
## 2014-10-01 2014-10-01    5.7
## 2014-11-01 2014-11-01    5.8
## 2014-12-01 2014-12-01    5.6
## 2015-01-01 2015-01-01    5.7
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
##                  date unrate vacrate
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
## 
## Call:
## lm(formula = unrate ~ vacrate, data = data.0)
## 
## Coefficients:
## (Intercept)      vacrate  
##       15.82        -3.51  
## 
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1        5.987320 4.567808 7.406832 3.808399 8.166241
## 2        5.636328 4.215735 7.056922 3.455747 7.816910
## 3        5.636328 4.215735 7.056922 3.455747 7.816910
## 4        5.987320 4.567808 7.406832 3.808399 8.166241
## 5        5.987320 4.567808 7.406832 3.808399 8.166241
## 6        5.636328 4.215735 7.056922 3.455747 7.816910
## 7        5.285337 3.863144 6.707530 3.102301 7.468373
## 8        4.934345 3.510036 6.358654 2.748061 7.120629
## 9        4.583353 3.156415 6.010291 2.393033 6.773673
## 10       4.232361 2.802282 5.662440 2.037220 6.427502
## 11       4.232361 2.802282 5.662440 2.037220 6.427502
## 12       3.881370 2.447642 5.315097 1.680628 6.082111
## 13       4.583353 3.156415 6.010291 2.393033 6.773673
## 14       3.881370 2.447642 5.315097 1.680628 6.082111
## 15       3.881370 2.447642 5.315097 1.680628 6.082111
## 16       3.881370 2.447642 5.315097 1.680628 6.082111
## 17       3.881370 2.447642 5.315097 1.680628 6.082111

##                  date unrate
## 2000-12-01 2000-12-01    3.9
## 2001-01-01 2001-01-01    4.2
## 2001-02-01 2001-02-01    4.2
## 2001-03-01 2001-03-01    4.3
## 2001-04-01 2001-04-01    4.4
## 2001-05-01 2001-05-01    4.3
## 2001-06-01 2001-06-01    4.5
## 2001-07-01 2001-07-01    4.6
## 2001-08-01 2001-08-01    4.9
## 2001-09-01 2001-09-01    5.0
## 2001-10-01 2001-10-01    5.3
## 2001-11-01 2001-11-01    5.5
## 2001-12-01 2001-12-01    5.7
## 2002-01-01 2002-01-01    5.7
## 2002-02-01 2002-02-01    5.7
## 2002-03-01 2002-03-01    5.7
## 2002-04-01 2002-04-01    5.9
## 2002-05-01 2002-05-01    5.8
## 2002-06-01 2002-06-01    5.8
## 2002-07-01 2002-07-01    5.8
## 2002-08-01 2002-08-01    5.7
## 2002-09-01 2002-09-01    5.7
## 2002-10-01 2002-10-01    5.7
## 2002-11-01 2002-11-01    5.9
## 2002-12-01 2002-12-01    6.0
## 2003-01-01 2003-01-01    5.8
## 2003-02-01 2003-02-01    5.9
## 2003-03-01 2003-03-01    5.9
## 2003-04-01 2003-04-01    6.0
## 2003-05-01 2003-05-01    6.1
## 2003-06-01 2003-06-01    6.3
## 2003-07-01 2003-07-01    6.2
## 2003-08-01 2003-08-01    6.1
## 2003-09-01 2003-09-01    6.1
## 2003-10-01 2003-10-01    6.0
## 2003-11-01 2003-11-01    5.8
## 2003-12-01 2003-12-01    5.7
## 2004-01-01 2004-01-01    5.7
## 2004-02-01 2004-02-01    5.6
## 2004-03-01 2004-03-01    5.8
## 2004-04-01 2004-04-01    5.6
## 2004-05-01 2004-05-01    5.6
## 2004-06-01 2004-06-01    5.6
## 2004-07-01 2004-07-01    5.5
## 2004-08-01 2004-08-01    5.4
## 2004-09-01 2004-09-01    5.4
## 2004-10-01 2004-10-01    5.5
## 2004-11-01 2004-11-01    5.4
## 2004-12-01 2004-12-01    5.4
## 2005-01-01 2005-01-01    5.3
## 2005-02-01 2005-02-01    5.4
## 2005-03-01 2005-03-01    5.2
## 2005-04-01 2005-04-01    5.2
## 2005-05-01 2005-05-01    5.1
## 2005-06-01 2005-06-01    5.0
## 2005-07-01 2005-07-01    5.0
## 2005-08-01 2005-08-01    4.9
## 2005-09-01 2005-09-01    5.0
## 2005-10-01 2005-10-01    5.0
## 2005-11-01 2005-11-01    5.0
## 2005-12-01 2005-12-01    4.9
## 2006-01-01 2006-01-01    4.7
## 2006-02-01 2006-02-01    4.8
## 2006-03-01 2006-03-01    4.7
## 2006-04-01 2006-04-01    4.7
## 2006-05-01 2006-05-01    4.6
## 2006-06-01 2006-06-01    4.6
## 2006-07-01 2006-07-01    4.7
## 2006-08-01 2006-08-01    4.7
## 2006-09-01 2006-09-01    4.5
## 2006-10-01 2006-10-01    4.4
## 2006-11-01 2006-11-01    4.5
## 2006-12-01 2006-12-01    4.4
## 2007-01-01 2007-01-01    4.6
## 2007-02-01 2007-02-01    4.5
## 2007-03-01 2007-03-01    4.4
## 2007-04-01 2007-04-01    4.5
## 2007-05-01 2007-05-01    4.4
## 2007-06-01 2007-06-01    4.6
## 2007-07-01 2007-07-01    4.7
## 2007-08-01 2007-08-01    4.6
## 2007-09-01 2007-09-01    4.7
## 2007-10-01 2007-10-01    4.7
## 2007-11-01 2007-11-01    4.7
## 2007-12-01 2007-12-01    5.0
## 2008-01-01 2008-01-01    5.0
## 2008-02-01 2008-02-01    4.9
## 2008-03-01 2008-03-01    5.1
## 2008-04-01 2008-04-01    5.0
## 2008-05-01 2008-05-01    5.4
## 2008-06-01 2008-06-01    5.6
## 2008-07-01 2008-07-01    5.8
## 2008-08-01 2008-08-01    6.1
## 2008-09-01 2008-09-01    6.1
## 2008-10-01 2008-10-01    6.5
## 2008-11-01 2008-11-01    6.8
## 2008-12-01 2008-12-01    7.3
## 2009-01-01 2009-01-01    7.8
## 2009-02-01 2009-02-01    8.3
## 2009-03-01 2009-03-01    8.7
## 2009-04-01 2009-04-01    9.0
## 2009-05-01 2009-05-01    9.4
## 2009-06-01 2009-06-01    9.5
## 2009-07-01 2009-07-01    9.5
## 2009-08-01 2009-08-01    9.6
## 2009-09-01 2009-09-01    9.8
## 2009-10-01 2009-10-01   10.0
## 2009-11-01 2009-11-01    9.9
## 2009-12-01 2009-12-01    9.9
## 2010-01-01 2010-01-01    9.8
## 2010-02-01 2010-02-01    9.8
## 2010-03-01 2010-03-01    9.9
## 2010-04-01 2010-04-01    9.9
## 2010-05-01 2010-05-01    9.6
## 2010-06-01 2010-06-01    9.4
## 2010-07-01 2010-07-01    9.4
## 2010-08-01 2010-08-01    9.5
## 2010-09-01 2010-09-01    9.5
## 2010-10-01 2010-10-01    9.4
## 2010-11-01 2010-11-01    9.8
## 2010-12-01 2010-12-01    9.3
## 2011-01-01 2011-01-01    9.2
## 2011-02-01 2011-02-01    9.0
## 2011-03-01 2011-03-01    9.0
## 2011-04-01 2011-04-01    9.1
## 2011-05-01 2011-05-01    9.0
## 2011-06-01 2011-06-01    9.1
## 2011-07-01 2011-07-01    9.0
## 2011-08-01 2011-08-01    9.0
## 2011-09-01 2011-09-01    9.0
## 2011-10-01 2011-10-01    8.8
## 2011-11-01 2011-11-01    8.6
## 2011-12-01 2011-12-01    8.5
## 2012-01-01 2012-01-01    8.3
## 2012-02-01 2012-02-01    8.3
## 2012-03-01 2012-03-01    8.2
## 2012-04-01 2012-04-01    8.2
## 2012-05-01 2012-05-01    8.2
## 2012-06-01 2012-06-01    8.2
## 2012-07-01 2012-07-01    8.2
## 2012-08-01 2012-08-01    8.0
## 2012-09-01 2012-09-01    7.8
## 2012-10-01 2012-10-01    7.8
## 2012-11-01 2012-11-01    7.7
## 2012-12-01 2012-12-01    7.9
## 2013-01-01 2013-01-01    8.0
## 2013-02-01 2013-02-01    7.7
## 2013-03-01 2013-03-01    7.5
## 2013-04-01 2013-04-01    7.6
## 2013-05-01 2013-05-01    7.5
## 2013-06-01 2013-06-01    7.5
## 2013-07-01 2013-07-01    7.3
## 2013-08-01 2013-08-01    7.2
## 2013-09-01 2013-09-01    7.2
## 2013-10-01 2013-10-01    7.2
## 2013-11-01 2013-11-01    7.0
## 2013-12-01 2013-12-01    6.7
## 2014-01-01 2014-01-01    6.6
## 2014-02-01 2014-02-01    6.7
## 2014-03-01 2014-03-01    6.6
## 2014-04-01 2014-04-01    6.2
## 2014-05-01 2014-05-01    6.3
## 2014-06-01 2014-06-01    6.1
## 2014-07-01 2014-07-01    6.2
## 2014-08-01 2014-08-01    6.1
## 2014-09-01 2014-09-01    5.9
## 2014-10-01 2014-10-01    5.7
## 2014-11-01 2014-11-01    5.8
## 2014-12-01 2014-12-01    5.6
## 2015-01-01 2015-01-01    5.7
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
##                  date unrate vacrate
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
## 
## Call:
## lm(formula = unrate ~ vacrate, data = data.0)
## 
## Coefficients:
## (Intercept)      vacrate  
##      15.608       -3.546  
## 
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1        7.452254 6.254387 8.650122 5.612208 9.292301
## 2        6.388509 5.193158 7.583860 4.552327 8.224690
## 3        6.743091 5.547344 7.938837 4.906302 8.579879
## 4        7.097672 5.901087 8.294258 5.259595 8.935750
## 5        6.388509 5.193158 7.583860 4.552327 8.224690
## 6        6.033927 4.838527 7.229326 4.197671 7.870183
## 7        6.388509 5.193158 7.583860 4.552327 8.224690
## 8        5.679345 4.483453 6.875237 3.842332 7.516357
## 9        6.388509 5.193158 7.583860 4.552327 8.224690
## 10       6.033927 4.838527 7.229326 4.197671 7.870183
## 11       6.033927 4.838527 7.229326 4.197671 7.870183
## 12       6.388509 5.193158 7.583860 4.552327 8.224690
## 13       6.388509 5.193158 7.583860 4.552327 8.224690
## 14       6.388509 5.193158 7.583860 4.552327 8.224690
## 15       6.033927 4.838527 7.229326 4.197671 7.870183
## 16       6.033927 4.838527 7.229326 4.197671 7.870183
## 17       6.388509 5.193158 7.583860 4.552327 8.224690
## 18       6.033927 4.838527 7.229326 4.197671 7.870183
## 19       5.324763 4.127935 6.521591 3.486312 7.163213
## 20       5.679345 4.483453 6.875237 3.842332 7.516357
## 21       6.033927 4.838527 7.229326 4.197671 7.870183
## 22       6.033927 4.838527 7.229326 4.197671 7.870183
## 23       5.679345 4.483453 6.875237 3.842332 7.516357
## 24       6.033927 4.838527 7.229326 4.197671 7.870183
## 25       5.679345 4.483453 6.875237 3.842332 7.516357
## 26       5.679345 4.483453 6.875237 3.842332 7.516357
## 27       5.324763 4.127935 6.521591 3.486312 7.163213
## 28       5.324763 4.127935 6.521591 3.486312 7.163213
## 29       5.679345 4.483453 6.875237 3.842332 7.516357
## 30       5.679345 4.483453 6.875237 3.842332 7.516357
## 31       5.324763 4.127935 6.521591 3.486312 7.163213
## 32       4.970181 3.771974 6.168387 3.129613 6.810749
## 33       4.615599 3.415573 5.815625 2.772236 6.458961
## 34       4.261017 3.058733 5.463301 2.414185 6.107848
## 35       3.906435 2.701456 5.111414 2.055464 5.757406
## 36       3.906435 2.701456 5.111414 2.055464 5.757406
## 37       3.551853 2.343746 4.759960 1.696077 5.407629
## 38       4.261017 3.058733 5.463301 2.414185 6.107848
## 39       3.551853 2.343746 4.759960 1.696077 5.407629
## 40       3.551853 2.343746 4.759960 1.696077 5.407629
## 41       3.551853 2.343746 4.759960 1.696077 5.407629
## 42       3.551853 2.343746 4.759960 1.696077 5.407629

##                  date unrate
## 2000-12-01 2000-12-01    3.9
## 2001-01-01 2001-01-01    4.2
## 2001-02-01 2001-02-01    4.2
## 2001-03-01 2001-03-01    4.3
## 2001-04-01 2001-04-01    4.4
## 2001-05-01 2001-05-01    4.3
## 2001-06-01 2001-06-01    4.5
## 2001-07-01 2001-07-01    4.6
## 2001-08-01 2001-08-01    4.9
## 2001-09-01 2001-09-01    5.0
## 2001-10-01 2001-10-01    5.3
## 2001-11-01 2001-11-01    5.5
## 2001-12-01 2001-12-01    5.7
## 2002-01-01 2002-01-01    5.7
## 2002-02-01 2002-02-01    5.7
## 2002-03-01 2002-03-01    5.7
## 2002-04-01 2002-04-01    5.9
## 2002-05-01 2002-05-01    5.8
## 2002-06-01 2002-06-01    5.8
## 2002-07-01 2002-07-01    5.8
## 2002-08-01 2002-08-01    5.7
## 2002-09-01 2002-09-01    5.7
## 2002-10-01 2002-10-01    5.7
## 2002-11-01 2002-11-01    5.9
## 2002-12-01 2002-12-01    6.0
## 2003-01-01 2003-01-01    5.8
## 2003-02-01 2003-02-01    5.9
## 2003-03-01 2003-03-01    5.9
## 2003-04-01 2003-04-01    6.0
## 2003-05-01 2003-05-01    6.1
## 2003-06-01 2003-06-01    6.3
## 2003-07-01 2003-07-01    6.2
## 2003-08-01 2003-08-01    6.1
## 2003-09-01 2003-09-01    6.1
## 2003-10-01 2003-10-01    6.0
## 2003-11-01 2003-11-01    5.8
## 2003-12-01 2003-12-01    5.7
## 2004-01-01 2004-01-01    5.7
## 2004-02-01 2004-02-01    5.6
## 2004-03-01 2004-03-01    5.8
## 2004-04-01 2004-04-01    5.6
## 2004-05-01 2004-05-01    5.6
## 2004-06-01 2004-06-01    5.6
## 2004-07-01 2004-07-01    5.5
## 2004-08-01 2004-08-01    5.4
## 2004-09-01 2004-09-01    5.4
## 2004-10-01 2004-10-01    5.5
## 2004-11-01 2004-11-01    5.4
## 2004-12-01 2004-12-01    5.4
## 2005-01-01 2005-01-01    5.3
## 2005-02-01 2005-02-01    5.4
## 2005-03-01 2005-03-01    5.2
## 2005-04-01 2005-04-01    5.2
## 2005-05-01 2005-05-01    5.1
## 2005-06-01 2005-06-01    5.0
## 2005-07-01 2005-07-01    5.0
## 2005-08-01 2005-08-01    4.9
## 2005-09-01 2005-09-01    5.0
## 2005-10-01 2005-10-01    5.0
## 2005-11-01 2005-11-01    5.0
## 2005-12-01 2005-12-01    4.9
## 2006-01-01 2006-01-01    4.7
## 2006-02-01 2006-02-01    4.8
## 2006-03-01 2006-03-01    4.7
## 2006-04-01 2006-04-01    4.7
## 2006-05-01 2006-05-01    4.6
## 2006-06-01 2006-06-01    4.6
## 2006-07-01 2006-07-01    4.7
## 2006-08-01 2006-08-01    4.7
## 2006-09-01 2006-09-01    4.5
## 2006-10-01 2006-10-01    4.4
## 2006-11-01 2006-11-01    4.5
## 2006-12-01 2006-12-01    4.4
## 2007-01-01 2007-01-01    4.6
## 2007-02-01 2007-02-01    4.5
## 2007-03-01 2007-03-01    4.4
## 2007-04-01 2007-04-01    4.5
## 2007-05-01 2007-05-01    4.4
## 2007-06-01 2007-06-01    4.6
## 2007-07-01 2007-07-01    4.7
## 2007-08-01 2007-08-01    4.6
## 2007-09-01 2007-09-01    4.7
## 2007-10-01 2007-10-01    4.7
## 2007-11-01 2007-11-01    4.7
## 2007-12-01 2007-12-01    5.0
## 2008-01-01 2008-01-01    5.0
## 2008-02-01 2008-02-01    4.9
## 2008-03-01 2008-03-01    5.1
## 2008-04-01 2008-04-01    5.0
## 2008-05-01 2008-05-01    5.4
## 2008-06-01 2008-06-01    5.6
## 2008-07-01 2008-07-01    5.8
## 2008-08-01 2008-08-01    6.1
## 2008-09-01 2008-09-01    6.1
## 2008-10-01 2008-10-01    6.5
## 2008-11-01 2008-11-01    6.8
## 2008-12-01 2008-12-01    7.3
## 2009-01-01 2009-01-01    7.8
## 2009-02-01 2009-02-01    8.3
## 2009-03-01 2009-03-01    8.7
## 2009-04-01 2009-04-01    9.0
## 2009-05-01 2009-05-01    9.4
## 2009-06-01 2009-06-01    9.5
## 2009-07-01 2009-07-01    9.5
## 2009-08-01 2009-08-01    9.6
## 2009-09-01 2009-09-01    9.8
## 2009-10-01 2009-10-01   10.0
## 2009-11-01 2009-11-01    9.9
## 2009-12-01 2009-12-01    9.9
## 2010-01-01 2010-01-01    9.8
## 2010-02-01 2010-02-01    9.8
## 2010-03-01 2010-03-01    9.9
## 2010-04-01 2010-04-01    9.9
## 2010-05-01 2010-05-01    9.6
## 2010-06-01 2010-06-01    9.4
## 2010-07-01 2010-07-01    9.4
## 2010-08-01 2010-08-01    9.5
## 2010-09-01 2010-09-01    9.5
## 2010-10-01 2010-10-01    9.4
## 2010-11-01 2010-11-01    9.8
## 2010-12-01 2010-12-01    9.3
## 2011-01-01 2011-01-01    9.2
## 2011-02-01 2011-02-01    9.0
## 2011-03-01 2011-03-01    9.0
## 2011-04-01 2011-04-01    9.1
## 2011-05-01 2011-05-01    9.0
## 2011-06-01 2011-06-01    9.1
## 2011-07-01 2011-07-01    9.0
## 2011-08-01 2011-08-01    9.0
## 2011-09-01 2011-09-01    9.0
## 2011-10-01 2011-10-01    8.8
## 2011-11-01 2011-11-01    8.6
## 2011-12-01 2011-12-01    8.5
## 2012-01-01 2012-01-01    8.3
## 2012-02-01 2012-02-01    8.3
## 2012-03-01 2012-03-01    8.2
## 2012-04-01 2012-04-01    8.2
## 2012-05-01 2012-05-01    8.2
## 2012-06-01 2012-06-01    8.2
## 2012-07-01 2012-07-01    8.2
## 2012-08-01 2012-08-01    8.0
## 2012-09-01 2012-09-01    7.8
## 2012-10-01 2012-10-01    7.8
## 2012-11-01 2012-11-01    7.7
## 2012-12-01 2012-12-01    7.9
## 2013-01-01 2013-01-01    8.0
## 2013-02-01 2013-02-01    7.7
## 2013-03-01 2013-03-01    7.5
## 2013-04-01 2013-04-01    7.6
## 2013-05-01 2013-05-01    7.5
## 2013-06-01 2013-06-01    7.5
## 2013-07-01 2013-07-01    7.3
## 2013-08-01 2013-08-01    7.2
## 2013-09-01 2013-09-01    7.2
## 2013-10-01 2013-10-01    7.2
## 2013-11-01 2013-11-01    7.0
## 2013-12-01 2013-12-01    6.7
## 2014-01-01 2014-01-01    6.6
## 2014-02-01 2014-02-01    6.7
## 2014-03-01 2014-03-01    6.6
## 2014-04-01 2014-04-01    6.2
## 2014-05-01 2014-05-01    6.3
## 2014-06-01 2014-06-01    6.1
## 2014-07-01 2014-07-01    6.2
## 2014-08-01 2014-08-01    6.1
## 2014-09-01 2014-09-01    5.9
## 2014-10-01 2014-10-01    5.7
## 2014-11-01 2014-11-01    5.8
## 2014-12-01 2014-12-01    5.6
## 2015-01-01 2015-01-01    5.7
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
##                  date unrate vacrate
## 2000-12-01 2000-12-01    3.9     3.6
## 2001-01-01 2001-01-01    4.2     3.8
## 2001-02-01 2001-02-01    4.2     3.4
## 2001-03-01 2001-03-01    4.3     3.4
## 2001-04-01 2001-04-01    4.4     3.4
## 2001-05-01 2001-05-01    4.3     3.2
## 2001-06-01 2001-06-01    4.5     3.1
## 2001-07-01 2001-07-01    4.6     3.2
## 2001-08-01 2001-08-01    4.9     3.0
## 2001-09-01 2001-09-01    5.0     2.9
## 2001-10-01 2001-10-01    5.3     2.7
## 2001-11-01 2001-11-01    5.5     2.6
## 2001-12-01 2001-12-01    5.7     2.7
## 2002-01-01 2002-01-01    5.7     2.7
## 2002-02-01 2002-02-01    5.7     2.5
## 2002-03-01 2002-03-01    5.7     2.7
## 2002-04-01 2002-04-01    5.9     2.5
## 2002-05-01 2002-05-01    5.8     2.6
## 2002-06-01 2002-06-01    5.8     2.5
## 2002-07-01 2002-07-01    5.8     2.5
## 2002-08-01 2002-08-01    5.7     2.6
## 2002-09-01 2002-09-01    5.7     2.5
## 2002-10-01 2002-10-01    5.7     2.6
## 2002-11-01 2002-11-01    5.9     2.7
## 2002-12-01 2002-12-01    6.0     2.3
## 2003-01-01 2003-01-01    5.8     2.7
## 2003-02-01 2003-02-01    5.9     2.5
## 2003-03-01 2003-03-01    5.9     2.3
## 2003-04-01 2003-04-01    6.0     2.4
## 2003-05-01 2003-05-01    6.1     2.4
## 2003-06-01 2003-06-01    6.3     2.5
## 2003-07-01 2003-07-01    6.2     2.4
## 2003-08-01 2003-08-01    6.1     2.4
## 2003-09-01 2003-09-01    6.1     2.3
## 2003-10-01 2003-10-01    6.0     2.4
## 2003-11-01 2003-11-01    5.8     2.5
## 2003-12-01 2003-12-01    5.7     2.5
## 2004-01-01 2004-01-01    5.7     2.5
## 2004-02-01 2004-02-01    5.6     2.6
## 2004-03-01 2004-03-01    5.8     2.5
## 2004-04-01 2004-04-01    5.6     2.6
## 2004-05-01 2004-05-01    5.6     2.7
## 2004-06-01 2004-06-01    5.6     2.5
## 2004-07-01 2004-07-01    5.5     2.8
## 2004-08-01 2004-08-01    5.4     2.6
## 2004-09-01 2004-09-01    5.4     2.7
## 2004-10-01 2004-10-01    5.5     2.8
## 2004-11-01 2004-11-01    5.4     2.5
## 2004-12-01 2004-12-01    5.4     2.8
## 2005-01-01 2005-01-01    5.3     2.6
## 2005-02-01 2005-02-01    5.4     2.8
## 2005-03-01 2005-03-01    5.2     2.8
## 2005-04-01 2005-04-01    5.2     3.0
## 2005-05-01 2005-05-01    5.1     2.8
## 2005-06-01 2005-06-01    5.0     2.9
## 2005-07-01 2005-07-01    5.0     3.0
## 2005-08-01 2005-08-01    4.9     2.9
## 2005-09-01 2005-09-01    5.0     3.0
## 2005-10-01 2005-10-01    5.0     3.0
## 2005-11-01 2005-11-01    5.0     3.1
## 2005-12-01 2005-12-01    4.9     3.0
## 2006-01-01 2006-01-01    4.7     3.1
## 2006-02-01 2006-02-01    4.8     3.1
## 2006-03-01 2006-03-01    4.7     3.2
## 2006-04-01 2006-04-01    4.7     3.2
## 2006-05-01 2006-05-01    4.6     3.2
## 2006-06-01 2006-06-01    4.6     3.1
## 2006-07-01 2006-07-01    4.7     2.9
## 2006-08-01 2006-08-01    4.7     3.1
## 2006-09-01 2006-09-01    4.5     3.2
## 2006-10-01 2006-10-01    4.4     3.2
## 2006-11-01 2006-11-01    4.5     3.3
## 2006-12-01 2006-12-01    4.4     3.2
## 2007-01-01 2007-01-01    4.6     3.2
## 2007-02-01 2007-02-01    4.5     3.2
## 2007-03-01 2007-03-01    4.4     3.3
## 2007-04-01 2007-04-01    4.5     3.2
## 2007-05-01 2007-05-01    4.4     3.2
## 2007-06-01 2007-06-01    4.6     3.3
## 2007-07-01 2007-07-01    4.7     3.1
## 2007-08-01 2007-08-01    4.6     3.2
## 2007-09-01 2007-09-01    4.7     3.1
## 2007-10-01 2007-10-01    4.7     3.0
## 2007-11-01 2007-11-01    4.7     3.0
## 2007-12-01 2007-12-01    5.0     3.0
##                  date unrate vacrate
## 2008-01-01 2008-01-01    5.0     3.0
## 2008-02-01 2008-02-01    4.9     2.8
## 2008-03-01 2008-03-01    5.1     2.8
## 2008-04-01 2008-04-01    5.0     2.7
## 2008-05-01 2008-05-01    5.4     2.8
## 2008-06-01 2008-06-01    5.6     2.6
## 2008-07-01 2008-07-01    5.8     2.7
## 2008-08-01 2008-08-01    6.1     2.5
## 2008-09-01 2008-09-01    6.1     2.3
## 2008-10-01 2008-10-01    6.5     2.3
## 2008-11-01 2008-11-01    6.8     2.2
## 2008-12-01 2008-12-01    7.3     2.2
## 2009-01-01 2009-01-01    7.8     2.0
## 2009-02-01 2009-02-01    8.3     2.1
## 2009-03-01 2009-03-01    8.7     1.8
## 2009-04-01 2009-04-01    9.0     1.7
## 2009-05-01 2009-05-01    9.4     1.8
## 2009-06-01 2009-06-01    9.5     1.8
## 2009-07-01 2009-07-01    9.5     1.6
## 2009-08-01 2009-08-01    9.6     1.7
## 2009-09-01 2009-09-01    9.8     1.8
## 2009-10-01 2009-10-01   10.0     1.8
## 2009-11-01 2009-11-01    9.9     1.8
## 2009-12-01 2009-12-01    9.9     1.9
## 2010-01-01 2010-01-01    9.8     2.0
## 2010-02-01 2010-02-01    9.8     1.9
## 2010-03-01 2010-03-01    9.9     2.0
## 2010-04-01 2010-04-01    9.9     2.3
## 2010-05-01 2010-05-01    9.6     2.2
## 2010-06-01 2010-06-01    9.4     2.0
## 2010-07-01 2010-07-01    9.4     2.2
## 2010-08-01 2010-08-01    9.5     2.2
## 2010-09-01 2010-09-01    9.5     2.1
## 2010-10-01 2010-10-01    9.4     2.3
## 2010-11-01 2010-11-01    9.8     2.3
## 2010-12-01 2010-12-01    9.3     2.2
## 2011-01-01 2011-01-01    9.2     2.2
## 2011-02-01 2011-02-01    9.0     2.3
## 2011-03-01 2011-03-01    9.0     2.3
## 2011-04-01 2011-04-01    9.1     2.3
## 2011-05-01 2011-05-01    9.0     2.3
## 2011-06-01 2011-06-01    9.1     2.4
## 2011-07-01 2011-07-01    9.0     2.5
## 2011-08-01 2011-08-01    9.0     2.3
## 2011-09-01 2011-09-01    9.0     2.6
## 2011-10-01 2011-10-01    8.8     2.5
## 2011-11-01 2011-11-01    8.6     2.4
## 2011-12-01 2011-12-01    8.5     2.6
## 2012-01-01 2012-01-01    8.3     2.7
## 2012-02-01 2012-02-01    8.3     2.6
## 2012-03-01 2012-03-01    8.2     2.8
## 2012-04-01 2012-04-01    8.2     2.6
## 2012-05-01 2012-05-01    8.2     2.7
## 2012-06-01 2012-06-01    8.2     2.7
## 2012-07-01 2012-07-01    8.2     2.6
## 2012-08-01 2012-08-01    8.0     2.6
## 2012-09-01 2012-09-01    7.8     2.6
## 2012-10-01 2012-10-01    7.8     2.7
## 2012-11-01 2012-11-01    7.7     2.7
## 2012-12-01 2012-12-01    7.9     2.6
## 2013-01-01 2013-01-01    8.0     2.7
## 2013-02-01 2013-02-01    7.7     2.9
## 2013-03-01 2013-03-01    7.5     2.8
## 2013-04-01 2013-04-01    7.6     2.7
## 2013-05-01 2013-05-01    7.5     2.7
## 2013-06-01 2013-06-01    7.5     2.8
## 2013-07-01 2013-07-01    7.3     2.7
## 2013-08-01 2013-08-01    7.2     2.8
## 2013-09-01 2013-09-01    7.2     2.8
## 2013-10-01 2013-10-01    7.2     2.9
## 2013-11-01 2013-11-01    7.0     2.9
## 2013-12-01 2013-12-01    6.7     2.8
## 2014-01-01 2014-01-01    6.6     2.8
## 2014-02-01 2014-02-01    6.7     2.9
## 2014-03-01 2014-03-01    6.6     3.0
## 2014-04-01 2014-04-01    6.2     3.1
## 2014-05-01 2014-05-01    6.3     3.2
## 2014-06-01 2014-06-01    6.1     3.3
## 2014-07-01 2014-07-01    6.2     3.3
## 2014-08-01 2014-08-01    6.1     3.4
## 2014-09-01 2014-09-01    5.9     3.2
## 2014-10-01 2014-10-01    5.7     3.4
## 2014-11-01 2014-11-01    5.8     3.4
## 2014-12-01 2014-12-01    5.6     3.4
## 2015-01-01 2015-01-01    5.7     3.4
## 
## Call:
## lm(formula = unrate ~ vacrate, data = data.0)
## 
## Coefficients:
## (Intercept)      vacrate  
##      10.022       -1.693  
## 
##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 1        4.942957 4.698686 5.187227 4.566868 5.319045
## 2        5.281590 5.037523 5.525657 4.905815 5.657365
## 3        5.281590 5.037523 5.525657 4.905815 5.657365
## 4        5.450907 5.206560 5.695253 5.074701 5.827112
## 5        5.281590 5.037523 5.525657 4.905815 5.657365
## 6        5.620224 5.375344 5.865103 5.243198 5.997249
## 7        5.450907 5.206560 5.695253 5.074701 5.827112
## 8        5.789540 5.543877 6.035204 5.411307 6.167774
## 9        6.128174 5.880197 6.376151 5.746379 6.509969
## 10       6.128174 5.880197 6.376151 5.746379 6.509969
## 11       6.297491 6.047992 6.546989 5.913353 6.681628
## 12       6.297491 6.047992 6.546989 5.913353 6.681628
## 13       6.636124 6.382873 6.889375 6.246209 7.026039
## 14       6.466807 6.215549 6.718066 6.079960 6.853655
## 15       6.974757 6.716845 7.232670 6.577665 7.371850
## 16       7.144074 6.883504 7.404644 6.742891 7.545258
## 17       6.974757 6.716845 7.232670 6.577665 7.371850
## 18       6.974757 6.716845 7.232670 6.577665 7.371850
## 19       7.313391 7.049955 7.576827 6.907795 7.718987
## 20       7.144074 6.883504 7.404644 6.742891 7.545258
## 21       6.974757 6.716845 7.232670 6.577665 7.371850
## 22       6.974757 6.716845 7.232670 6.577665 7.371850
## 23       6.974757 6.716845 7.232670 6.577665 7.371850
## 24       6.805441 6.549970 7.060912 6.412107 7.198774
## 25       6.636124 6.382873 6.889375 6.246209 7.026039
## 26       6.805441 6.549970 7.060912 6.412107 7.198774
## 27       6.636124 6.382873 6.889375 6.246209 7.026039
## 28       6.128174 5.880197 6.376151 5.746379 6.509969
## 29       6.297491 6.047992 6.546989 5.913353 6.681628
## 30       6.636124 6.382873 6.889375 6.246209 7.026039
## 31       6.297491 6.047992 6.546989 5.913353 6.681628
## 32       6.297491 6.047992 6.546989 5.913353 6.681628
## 33       6.466807 6.215549 6.718066 6.079960 6.853655
## 34       6.128174 5.880197 6.376151 5.746379 6.509969
## 35       6.128174 5.880197 6.376151 5.746379 6.509969
## 36       6.297491 6.047992 6.546989 5.913353 6.681628
## 37       6.297491 6.047992 6.546989 5.913353 6.681628
## 38       6.128174 5.880197 6.376151 5.746379 6.509969
## 39       6.128174 5.880197 6.376151 5.746379 6.509969
## 40       6.128174 5.880197 6.376151 5.746379 6.509969
## 41       6.128174 5.880197 6.376151 5.746379 6.509969
## 42       5.958857 5.712160 6.205554 5.579032 6.338682
## 43       5.789540 5.543877 6.035204 5.411307 6.167774
## 44       6.128174 5.880197 6.376151 5.746379 6.509969
## 45       5.620224 5.375344 5.865103 5.243198 5.997249
## 46       5.789540 5.543877 6.035204 5.411307 6.167774
## 47       5.958857 5.712160 6.205554 5.579032 6.338682
## 48       5.620224 5.375344 5.865103 5.243198 5.997249
## 49       5.450907 5.206560 5.695253 5.074701 5.827112
## 50       5.620224 5.375344 5.865103 5.243198 5.997249
## 51       5.281590 5.037523 5.525657 4.905815 5.657365
## 52       5.620224 5.375344 5.865103 5.243198 5.997249
## 53       5.450907 5.206560 5.695253 5.074701 5.827112
## 54       5.450907 5.206560 5.695253 5.074701 5.827112
## 55       5.620224 5.375344 5.865103 5.243198 5.997249
## 56       5.620224 5.375344 5.865103 5.243198 5.997249
## 57       5.620224 5.375344 5.865103 5.243198 5.997249
## 58       5.450907 5.206560 5.695253 5.074701 5.827112
## 59       5.450907 5.206560 5.695253 5.074701 5.827112
## 60       5.620224 5.375344 5.865103 5.243198 5.997249
## 61       5.450907 5.206560 5.695253 5.074701 5.827112
## 62       5.112273 4.868232 5.356315 4.736537 5.488009
## 63       5.281590 5.037523 5.525657 4.905815 5.657365
## 64       5.450907 5.206560 5.695253 5.074701 5.827112
## 65       5.450907 5.206560 5.695253 5.074701 5.827112
## 66       5.281590 5.037523 5.525657 4.905815 5.657365
## 67       5.450907 5.206560 5.695253 5.074701 5.827112
## 68       5.281590 5.037523 5.525657 4.905815 5.657365
## 69       5.281590 5.037523 5.525657 4.905815 5.657365
## 70       5.112273 4.868232 5.356315 4.736537 5.488009
## 71       5.112273 4.868232 5.356315 4.736537 5.488009
## 72       5.281590 5.037523 5.525657 4.905815 5.657365
## 73       5.281590 5.037523 5.525657 4.905815 5.657365
## 74       5.112273 4.868232 5.356315 4.736537 5.488009
## 75       4.942957 4.698686 5.187227 4.566868 5.319045
## 76       4.773640 4.528887 5.018392 4.396809 5.150471
## 77       4.604323 4.358836 4.849810 4.226362 4.982284
## 78       4.435006 4.188536 4.681477 4.055530 4.814482
## 79       4.435006 4.188536 4.681477 4.055530 4.814482
## 80       4.265690 4.017988 4.513391 3.884319 4.647060
## 81       4.604323 4.358836 4.849810 4.226362 4.982284
## 82       4.265690 4.017988 4.513391 3.884319 4.647060
## 83       4.265690 4.017988 4.513391 3.884319 4.647060
## 84       4.265690 4.017988 4.513391 3.884319 4.647060
## 85       4.265690 4.017988 4.513391 3.884319 4.647060

4.8 Time-Series Specific Issues

Residual autocorrelation:
- Time series data tend to be persistent.
- Symptom may be autocorrelated residuals.
- Omitting lagged dependent variable will likely bias forecasts.
Spurious regression:
- Totally unrelated trending (non-stationary) data series can appear highly correlated.
- Still problematic for likely-related data series.
- Best practice: Plot data beforehand

5.1 Multiple Regression

General form of \(k\)-variable multiple regression: \[ y_{t} = \beta_{0} + \beta_{1} x_{1,t} + \beta_{2} x_{2,t} + \cdots + \beta_{k} x_{k,t} + e_{t}. \]
\(\beta_1,\dots,\beta_k\) measure partial effect of \(k^{th}\) regressor.
- Controlling for all other variables in model.
Estimation proceeds as before, with OLS minimising: \[ \sum_{i=1}^N e_{t}^2 = \sum_{i=1}^N (y_{t} - \beta_{0} - \beta_{1}x_{1,t} - \cdots - \beta_{k} x_{k,t})^2. \]

5.2 Some Useful Predictors

Distributed lag:
- Effect of \(x\) on \(y\) may be distributed through time.
- Include more than one lag of \(x\) in model.
Many deterministic (non-random) terms helpful for forecasting.
Dummy variables:
- Seasonal dummies: 3 quarterly dummies, 11 monthly dummies.
- Outliers: \((0,\dots,0,1,0,\dots,0)\).
- Shift dummies: e.g. \((0,\dots,0,1,\dots,1)\).
Trend terms: Time trend for trending variables.

5.3 Selecting Regressors

Graphical methods not recommended: Naked eye misses relationships.
Adjusted \(R^2\): \(R^2\) increases with every additional regressor.
- \(\overline{R}^2\) penalises extra regressors: \[ \bar{R}^2 = 1-(1-R^2)\frac{N-1}{N-k-1}, \]
Cross Validation:
- Remove observation \(i\) from regression sample. Compute error: \(e_{t}^*=y_{t}-\hat{y}_{t}\).
- Repeat for all observations, \(i=1,\dots,N\).
- Compute MSE for \(e_1^*,e_2^*,\dots,e_N^*\), call it \(CV\).
- Choose model with smallest \(CV\).

5.3 Selecting Regressors

Information criteria: Penalise function of \(SSE\).
- Select model based on smallest information criterion chosen.
- Akaike, Bayesian most common.
Stepwise regression: Model selection algorithm.
- Start with smallest/largest model.
- Add/remove regressor. Calculate measure (e.g. AIC). Keep if improved.
- Continue until no further improvement.
- Stepwise can yield bad results:
Much more next week…

5.4 Residual Diagnostics

Residuals: \(e_{t} = y_{t}-\hat{y}_{t}\). Health-check for model.
Scatterplot of residuals against regressors:
- Fundamental assumption: \(\E{}{\ols{\eps}_t\cond{x_t}}=0\).
Scatterplot of residuals against fitted values:
- Should be uncorrelated. Correlation implies heteroskedasticity.
Residual autocorrelation:
- Autoregression on residuals better than DW test.
Residual density: Use histogram, should look `normal’.

5.5 Non-Linear Regression

May want to do more than add non-linear regressors: \[ y = f(x) + e. \]
Can still do some things in linear regression model:
- E.g. split trend:

\[ z = (x-25)_{+} = \begin{cases} 0 & \text{if} x < 25 \\ x-25 & \text{if} x\ge 25 \end{cases} \] - May want yet more flexible trend: Regression spline. + Arbitrarily smoothed trend term. + However, unreliable for forecasting out-of-sample.

5.7 Correlation, Causation and Forecasting

Causality essential in regression modelling:
- We need to know change in \(y\) came from \(x\) not \(\eps\).
- Zero conditional mean assumption \(\E{\eps\cond{x}}=0\) ensures.
Causality helpful but not essential for forecasting.
E.g.: Drownings on beach, and ice creams sold.
- Political contributions and election outcomes…?
Warnings:
- Without causality, may be harder to explain forecasts.
- Multicollinearity is problem if regressors very highly correlated.