EC313 Business and Financial Forecasting Lecture 4
J James Reade
21/01/2015
Introduction
\[
\newcommand{\N}[1]{\mathsf{N}\left(#1\right)}
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\newcommand{\V}[1]{\mathsf{Var}\left(#1\right)}
\newcommand{\C}[1]{\mathsf{Cov}\left(#1\right)}
\newcommand{\ols}[1]{\widehat{#1}}
\newcommand{\cond}[1]{\left|#1\right.}
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\]
- Forecasting: Essential for life.
- Forecast the work required to get slides ready…
- Thus far:
- Judgemental forecasting.
- Linear regression.
- ARIMA.
- Methods to forecast.
- But how do we know our model is any good?
Today: Model Selection and Impulse Saturation
- Model selection:
- How good is our model?
- Need to select a model, but how?
- Impulse saturation:
- Need to account for outliers.
- Need to account for structural breaks.
Model Selection
- Vast number of potential models:
- With \(k\) explanatory variables, \(2^k\) possible models.
- Computationally implausible to estimate all.
- Nonetheless, how do we select model?
- Important considerations:
- Effect of misspecification.
- Omitted variable bias.
- Over-fitting.
- Alternatives:
- FPP: Stepwise
- Alternative: General-to-specific.
How Good is Our Model?
- How do we measure how good a model is?
- \(R^2\): \[
R^2 = \frac{\sum(\hat{y}_{i} - \bar{y})^2}{\sum(y_{i}-\bar{y})^2},
\]
- But \(R^2\) increases with \(k\).
- Regardless of the importance of additional regressors.
- We can correct for this problem in \(R^2\). Penalise \(k\).
- Basic idea: Estimation uncertainly \(\uparrow\) in \(k\).
- Example is adjusted-\(R^2\): \[
\bar{R}^2 = 1-(1-R^2)\frac{N-1}{N-k-1}.
\]
- Alternatives are information criteria:
- Akiaike: \(\text{AIC} = N\log\left(\frac{\text{SSE}}{N}\right)+2(k+2),\)
- Corrected Akiaike: \(\text{AIC}_{\text{c}} = \text{AIC} + \frac{2(k+2)(k+3)}{N-k-3}.\)
- Corrected for small-sample problems in AIC.
- Bayesian/Schwarz: \(\text{BIC} = N\log\left(\frac{\text{SSE}}{N}\right) + (k+2)\log(N).\)
- Cross-validation:
- Remove observation \(s\) from sample and estimate model.
- Compute some measure based on residuals (e.g. MSE).
- Repeat for all \(T\) and create CV series \(e^*_1,\dots,e^*_T\).
- Model with smallest CV series is best.
Misspecification
- Recall basic (in-sample) model: \[
y_t = \beta_0 + \beta_1 x_t + e_t, \qquad e_t\sim iid(0,\sigma^2).
\]
- Important assumptions:
- \(\E{e_t\cond{x_t}}=0\).
- \(\V{e_t\cond{x_t}}=\sigma^2_e\).
- \(\C{e_t,e_s\cond{x_t,x_s}}=0\), for \(t\neq s\).
- Imply residuals look like:
xn <- rnorm(1000000)
hist(xn,freq=F,main="iid residual distribution")
lines(density(xn),col=2)
Implications of Violations of iidness
- Autocorrelation: \(\C{e_t,e_s\cond{x_t,x_s}}\neq0\), for some \(t\neq s\).
- Undoubtedly bias via omitted variable bias.
- Autocorrelation implies systematic information omitted from model.
- Heteroskedasticity: \(\V{e_t\cond{x_t}}=\sigma^2_{e,t}\).
- Increased standard errors in regression model.
- Increased estimation uncertainty.
- Non-normality:
- Combination of the above.
- Misspecification tests rarely identify type of misspecification:
- Almost always related to important omitted variables.
- Wrong functional form also possibility.
Omitted Variable Bias
- True (DGP) is: \(y_t = \beta_0 + \beta_1 x_{1,t} + \beta_2 x_{2,t} + e_t\).
- We model \(y_t = \beta_0 + \beta_1 x_{1,t} + e_t\).
- Get estimator: \[
\ols{\beta}_1 = \frac{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)y_t}{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)^2}
\]
- Substitute in true \(y_t=\beta_0 + \beta_1 x_{1,t} + \beta_2 x_{2,t} + e_t\) to get: \[
\E{\ols{\beta}_1} = \beta_1 + \E{\frac{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)\left(x_{2,t}-\overline{x}_2\right)}{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)^2}}\beta_2 + \E{\frac{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)e_t}{\sum_{t=1}^T \left(x_{1,t}-\overline{x}_1\right)^2}}
\]
- Evaluating RHS terms gives: \[
\E{\ols{\beta}_1} = \beta_1 + \frac{\C{x_{1,t},x_{2,t}}}{\V{x_{1,t}}}\beta_2 + \frac{\C{x_{1,t},e_{t}}}{\V{x_{1,t}}}.
\]
- iidness ensures \(\C{x_{1,t},e_{t}}=0\), but \(\C{x_{1,t},x_{2,t}}\)?
- If omitted variable related to anything in our model, we get bias.
- Implication: We need to include everything that matters!
- In practice: How much can we really include?
- Often, lagged dependent variables act as proxies.
- Suggests role for ARIMA ‘residuals’.
Over-Fitting
- Alternative situation: Too many variables.
- True (DGP) is: \(y_t = \beta_0 + \beta_1 x_{1,t} + e_t\).
- We model \(y_t = \beta_0 + \beta_1 x_{1,t} + \beta_2 x_{2,t} + e_t\).
- Hence irrelevant \(x_{2,t}\) included, hence \(\E{\ols{\beta}_2}=0\).
- Resulting \(\ols{\beta}_1\) estimator must still be unbiased.
- But Gauss-Markov tells us standard error of \(\ols{\beta}_1\) larger.
Over-Fitting
- More subtle problem: \(\beta_2\) may adventitiously capture variation in \(y_t\).
- We work with expected values; cannot rule out freak occurrences.
- Recall problem with \(R^2\):
- Increase number of regressors \(k\), increase \(R^2\).
library(zoo)
##
## Attaching package: 'zoo'
##
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
library(gets)
library(quantmod)
## Loading required package: xts
## Loading required package: TTR
## Version 0.4-0 included new data defaults. See ?getSymbols.
football <- read.csv("football.csv",stringsAsFactor=F)
football.z <- zooreg(football$x,order.by=football$Group.1)
dums <- data.frame("d1"=rep(0,NROW(football)))
dums[1,1] <- 1
for(i in 2:50) {
dums[[paste("d",i,sep="")]] <- rep(0,NROW(football))
dums[i,i] <- 1
}
dums <- zooreg(dums,order.by=football$Group.1)
football.arx <- arx(football.z,mxreg=dums)
Guarding Against Over-Fitting
- Try to include all important variables:
- Ensures iidness holds in residuals.
- Be aware of important omitted variables.
- Attempt to explain all obvious patterns in data.
Stepwise Regression
- Add or remove variables one-by-one.
- Backward stepwise:
- Start with model containing all variables.
- Remove one at a time, keep if smaller model increases some measure of predictive accuracy.
- Adjusted-\(R^2\), cross validation, AIC, AICc, BIC.
- Continue until no further improvement possible.
- Ideally order variables in terms of significance.
- Forward starts with smallest model and adds until no improvement.
Problems with Stepwise Regression
- Important: Most users aware of drawbacks of stepwise regression.
- FPP: “almost always leads to a good model”.
- Path dependency: Different paths, different models.
- Particularly in context of multicollinearity.
- Be aware of overly sanctimonious criticisms:
General-to-Specific
- Alternative: General-to-specific modelling.
- Associated with David F. Hendry and the ‘LSE Approach’.
- See http://goo.gl/pzrSG5 for overview (technical).
- Start with most general model possible:
- Referred to as General Unrestricted Model (GUM).
- Basic criterion: Model well specified.
- No autocorrelation, homoskedasticity, normal residuals.
- Reduce by removing variables only if:
- t- and F-tests allow variable removal.
- Smaller model satisfies misspecification test.
- Mitigate path dependency by searching multiple paths.
- No need to search all \(2^k\) models however.
- Seminal paper (worth reading):
Why Trust Selection?
- Numerous simulation studies in recent years.
- Simulate data according to data generating process.
- Important: We know what true model is.
- Apply selection algorithm and check selected model.
- Record how often true model selected.
- Better: Bias in estimated coefficients of final model.
Selection in R
xvars <- data.frame("v1"=rnorm(100))
for(i in 2:11) {
xvars[[paste("v",i,sep="")]] <- rnorm(100)
}
step.mod <- lm(v1 ~ v2 + v3 + v4 + v5 + v6 + v7 + v8 + v9 + v10,data=xvars)
search.step <- step(step.mod,direction="backward")
## Start: AIC=-12.01
## v1 ~ v2 + v3 + v4 + v5 + v6 + v7 + v8 + v9 + v10
##
## Df Sum of Sq RSS AIC
## - v5 1 0.00046 72.609 -14.008
## - v2 1 0.00235 72.611 -14.006
## - v9 1 0.01661 72.625 -13.986
## - v4 1 0.08884 72.697 -13.887
## - v3 1 0.19067 72.799 -13.747
## - v8 1 0.37555 72.984 -13.493
## - v10 1 0.66626 73.275 -13.095
## - v6 1 1.03925 73.648 -12.588
## - v7 1 1.11119 73.720 -12.490
## <none> 72.608 -12.009
##
## Step: AIC=-14.01
## v1 ~ v2 + v3 + v4 + v6 + v7 + v8 + v9 + v10
##
## Df Sum of Sq RSS AIC
## - v2 1 0.00206 72.611 -16.006
## - v9 1 0.01639 72.625 -15.986
## - v4 1 0.09061 72.699 -15.884
## - v3 1 0.19314 72.802 -15.743
## - v8 1 0.39362 73.002 -15.468
## - v10 1 0.66602 73.275 -15.095
## - v6 1 1.04244 73.651 -14.583
## - v7 1 1.11079 73.720 -14.490
## <none> 72.609 -14.008
##
## Step: AIC=-16.01
## v1 ~ v3 + v4 + v6 + v7 + v8 + v9 + v10
##
## Df Sum of Sq RSS AIC
## - v9 1 0.01712 72.628 -17.982
## - v4 1 0.08920 72.700 -17.883
## - v3 1 0.19241 72.803 -17.741
## - v8 1 0.39813 73.009 -17.459
## - v10 1 0.66396 73.275 -17.095
## - v6 1 1.04189 73.653 -16.581
## - v7 1 1.10913 73.720 -16.489
## <none> 72.611 -16.006
##
## Step: AIC=-17.98
## v1 ~ v3 + v4 + v6 + v7 + v8 + v10
##
## Df Sum of Sq RSS AIC
## - v4 1 0.09248 72.721 -19.855
## - v3 1 0.18847 72.817 -19.723
## - v8 1 0.38786 73.016 -19.449
## - v10 1 0.66221 73.290 -19.074
## - v6 1 1.02805 73.656 -18.576
## - v7 1 1.11772 73.746 -18.455
## <none> 72.628 -17.982
##
## Step: AIC=-19.85
## v1 ~ v3 + v6 + v7 + v8 + v10
##
## Df Sum of Sq RSS AIC
## - v3 1 0.14873 72.869 -21.650
## - v8 1 0.40909 73.130 -21.294
## - v10 1 0.61858 73.339 -21.008
## - v6 1 0.99414 73.715 -20.497
## - v7 1 1.12477 73.845 -20.320
## <none> 72.721 -19.855
##
## Step: AIC=-21.65
## v1 ~ v6 + v7 + v8 + v10
##
## Df Sum of Sq RSS AIC
## - v8 1 0.46438 73.334 -23.015
## - v10 1 0.56677 73.436 -22.876
## - v6 1 0.98053 73.850 -22.314
## - v7 1 1.02142 73.891 -22.258
## <none> 72.869 -21.650
##
## Step: AIC=-23.02
## v1 ~ v6 + v7 + v10
##
## Df Sum of Sq RSS AIC
## - v10 1 0.48092 73.815 -24.361
## - v6 1 0.81365 74.147 -23.912
## - v7 1 0.95016 74.284 -23.728
## <none> 73.334 -23.015
##
## Step: AIC=-24.36
## v1 ~ v6 + v7
##
## Df Sum of Sq RSS AIC
## - v7 1 0.87142 74.686 -25.188
## - v6 1 0.97214 74.787 -25.053
## <none> 73.815 -24.361
##
## Step: AIC=-25.19
## v1 ~ v6
##
## Df Sum of Sq RSS AIC
## - v6 1 1.0866 75.773 -25.743
## <none> 74.686 -25.188
##
## Step: AIC=-25.74
## v1 ~ 1
- General-to-specific (Gets):
library(gets)
x.z <- zooreg(xvars)
xvars.arx <- arx(x.z[,1],mxreg=x.z[,2:ncol(x.z)])
search.gets <- getsm(xvars.arx)
## Searching path no. 1 out of 10
## Searching path no. 2 out of 10
## Searching path no. 3 out of 10
## Searching path no. 4 out of 10
## Searching path no. 5 out of 10
## Searching path no. 6 out of 10
## Searching path no. 7 out of 10
## Searching path no. 8 out of 10
## Searching path no. 9 out of 10
## Searching path no. 10 out of 10
A Sample of One is Unhelpful
- Need many replications in simulations.
- Hoover and Perez (1999):
- various DGPs, 0–5 significant variables.
- GUM always has 40 variables.
- 75% of time true model in selected model.
- Numerous subsequent improvements in algorithm:
- Software variants:
- Autometrics, coded by Jurgen Doornik in OxMetrics.
- gets 0.2, coded by Genaro Sucarrat.
Example of Model Selection
- Beveridge Curve example: Vacancy rate on unemployment rate.
- But how many lags are necessary? Which variables?
- Might other variables matter?
- Matching process: Number of hires? Long-term unemployed?
- Participation rate?
- Frictional unemployment?
getSymbols('JTSJOR',src='FRED',return.class="zoo") #vacancy rate
## As of 0.4-0, 'getSymbols' uses env=parent.frame() and
## auto.assign=TRUE by default.
##
## This behavior will be phased out in 0.5-0 when the call will
## default to use auto.assign=FALSE. getOption("getSymbols.env") and
## getOptions("getSymbols.auto.assign") are now checked for alternate defaults
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbol for more details
## [1] "JTSJOR"
getSymbols('UNRATE',src='FRED',return.class="zoo") #unemployment rate
## [1] "UNRATE"
getSymbols('JTSQUR',src='FRED',return.class="zoo") #quits
## [1] "JTSQUR"
getSymbols('JTSHIL',src='FRED',return.class="zoo") #total hires
## [1] "JTSHIL"
getSymbols('LNU01300000',src='FRED',return.class="zoo") #participation rate
## [1] "LNU01300000"
getSymbols('UEMPMEAN',src='FRED',return.class="zoo") #average duration of unemployment
## [1] "UEMPMEAN"
Current and lag of every variable:
partrate <- window(LNU01300000, start="2000-12-01",end="2014-11-01")
uempmean <- window(UEMPMEAN, start="2000-12-01",end="2014-11-01")
unrate <- window(UNRATE, start="2000-12-01",end="2014-11-01")
vars <- data.frame(partrate,uempmean,unrate,JTSHIL,JTSJOR,JTSQUR,
"partrate.1"=lag(partrate,-1,na.pad=TRUE),
"uempmean.1"=lag(uempmean,-1,na.pad=TRUE),
"unrate.1"=lag(unrate,-1,na.pad=TRUE),
"JTSHIL.1"=lag(JTSHIL,-1,na.pad=TRUE),
"JTSQUR.1"=lag(JTSQUR,-1,na.pad=TRUE),
"JTSJOR.1"=lag(JTSJOR,-1,na.pad=TRUE))
vac.mod <- lm(JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate + partrate.1 +
JTSHIL + JTSHIL.1 + JTSQUR + JTSQUR.1 + uempmean + uempmean.1, data=vars)
summary(vac.mod)
##
## Call:
## lm(formula = JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate +
## partrate.1 + JTSHIL + JTSHIL.1 + JTSQUR + JTSQUR.1 + uempmean +
## uempmean.1, data = vars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.37599 -0.06788 0.00121 0.07611 0.33975
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.956e+00 1.361e+00 1.437 0.15284
## JTSJOR.1 5.867e-01 5.941e-02 9.875 < 2e-16 ***
## unrate 4.902e-02 6.950e-02 0.705 0.48166
## unrate.1 -7.346e-02 6.767e-02 -1.086 0.27938
## partrate -8.880e-03 3.317e-02 -0.268 0.78929
## partrate.1 -2.928e-02 3.350e-02 -0.874 0.38340
## JTSHIL 2.239e-04 7.866e-05 2.846 0.00503 **
## JTSHIL.1 3.353e-05 8.514e-05 0.394 0.69429
## JTSQUR -8.757e-02 1.540e-01 -0.569 0.57044
## JTSQUR.1 2.880e-01 1.455e-01 1.980 0.04950 *
## uempmean 5.101e-03 1.463e-02 0.349 0.72775
## uempmean.1 3.908e-03 1.386e-02 0.282 0.77839
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1198 on 155 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9241, Adjusted R-squared: 0.9187
## F-statistic: 171.6 on 11 and 155 DF, p-value: < 2.2e-16
Stepwise selection:
vac.step <- step(vac.mod)
## Start: AIC=-697.21
## JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate + partrate.1 +
## JTSHIL + JTSHIL.1 + JTSQUR + JTSQUR.1 + uempmean + uempmean.1
##
## Df Sum of Sq RSS AIC
## - partrate 1 0.00103 2.2252 -699.13
## - uempmean.1 1 0.00114 2.2253 -699.12
## - uempmean 1 0.00175 2.2259 -699.08
## - JTSHIL.1 1 0.00223 2.2264 -699.04
## - JTSQUR 1 0.00464 2.2288 -698.86
## - unrate 1 0.00714 2.2313 -698.67
## - partrate.1 1 0.01096 2.2352 -698.39
## - unrate.1 1 0.01691 2.2411 -697.94
## <none> 2.2242 -697.21
## - JTSQUR.1 1 0.05625 2.2804 -695.04
## - JTSHIL 1 0.11624 2.3404 -690.70
## - JTSJOR.1 1 1.39938 3.6236 -617.70
##
## Step: AIC=-699.13
## JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate.1 + JTSHIL +
## JTSHIL.1 + JTSQUR + JTSQUR.1 + uempmean + uempmean.1
##
## Df Sum of Sq RSS AIC
## - uempmean.1 1 0.00141 2.2266 -701.02
## - uempmean 1 0.00160 2.2268 -701.01
## - JTSHIL.1 1 0.00206 2.2273 -700.97
## - JTSQUR 1 0.00438 2.2296 -700.80
## - unrate 1 0.00640 2.2316 -700.65
## - unrate.1 1 0.01604 2.2413 -699.93
## <none> 2.2252 -699.13
## - partrate.1 1 0.04888 2.2741 -697.50
## - JTSQUR.1 1 0.05536 2.2806 -697.03
## - JTSHIL 1 0.11522 2.3404 -692.70
## - JTSJOR.1 1 1.41558 3.6408 -618.91
##
## Step: AIC=-701.02
## JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate.1 + JTSHIL +
## JTSHIL.1 + JTSQUR + JTSQUR.1 + uempmean
##
## Df Sum of Sq RSS AIC
## - JTSHIL.1 1 0.00290 2.2295 -702.81
## - JTSQUR 1 0.00507 2.2317 -702.64
## - unrate 1 0.00673 2.2334 -702.52
## - unrate.1 1 0.01694 2.2436 -701.76
## <none> 2.2266 -701.02
## - partrate.1 1 0.04957 2.2762 -699.35
## - JTSQUR.1 1 0.05531 2.2819 -698.93
## - uempmean 1 0.07260 2.2992 -697.67
## - JTSHIL 1 0.11416 2.3408 -694.67
## - JTSJOR.1 1 1.42323 3.6499 -620.49
##
## Step: AIC=-702.81
## JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate.1 + JTSHIL +
## JTSQUR + JTSQUR.1 + uempmean
##
## Df Sum of Sq RSS AIC
## - JTSQUR 1 0.00515 2.2347 -704.42
## - unrate 1 0.00523 2.2348 -704.42
## - unrate.1 1 0.01534 2.2449 -703.66
## <none> 2.2295 -702.81
## - partrate.1 1 0.04785 2.2774 -701.26
## - uempmean 1 0.07727 2.3068 -699.12
## - JTSQUR.1 1 0.09055 2.3201 -698.16
## - JTSHIL 1 0.13155 2.3611 -695.23
## - JTSJOR.1 1 1.45723 3.6868 -620.81
##
## Step: AIC=-704.42
## JTSJOR ~ JTSJOR.1 + unrate + unrate.1 + partrate.1 + JTSHIL +
## JTSQUR.1 + uempmean
##
## Df Sum of Sq RSS AIC
## - unrate 1 0.00588 2.2406 -705.98
## - unrate.1 1 0.01653 2.2512 -705.19
## <none> 2.2347 -704.42
## - partrate.1 1 0.05075 2.2854 -702.67
## - uempmean 1 0.08608 2.3208 -700.11
## - JTSQUR.1 1 0.08723 2.3219 -700.03
## - JTSHIL 1 0.15940 2.3941 -694.91
## - JTSJOR.1 1 1.49593 3.7306 -620.84
##
## Step: AIC=-705.98
## JTSJOR ~ JTSJOR.1 + unrate.1 + partrate.1 + JTSHIL + JTSQUR.1 +
## uempmean
##
## Df Sum of Sq RSS AIC
## - unrate.1 1 0.02498 2.2655 -706.13
## <none> 2.2406 -705.98
## - partrate.1 1 0.05091 2.2915 -704.23
## - uempmean 1 0.08069 2.3212 -702.07
## - JTSQUR.1 1 0.08177 2.3223 -702.00
## - JTSHIL 1 0.15623 2.3968 -696.73
## - JTSJOR.1 1 1.52274 3.7633 -621.38
##
## Step: AIC=-706.13
## JTSJOR ~ JTSJOR.1 + partrate.1 + JTSHIL + JTSQUR.1 + uempmean
##
## Df Sum of Sq RSS AIC
## <none> 2.2655 -706.13
## - uempmean 1 0.05675 2.3223 -704.00
## - partrate.1 1 0.08921 2.3547 -701.68
## - JTSQUR.1 1 0.11858 2.3841 -699.61
## - JTSHIL 1 0.24060 2.5061 -691.27
## - JTSJOR.1 1 2.01074 4.2763 -602.04
Gets selection:
vac.arx <- arx(vars$JTSJOR[2:NROW(vars)],mc=T,ar=1:5,
mxreg = cbind(vars$unrate[2:NROW(vars)],vars$unrate.1[2:NROW(vars)],
vars$partrate[2:NROW(vars)],vars$partrate.1[2:NROW(vars)],
vars$JTSHIL[2:NROW(vars)],vars$JTSHIL.1[2:NROW(vars)],
vars$JTSQUR[2:NROW(vars)],vars$JTSQUR.1[2:NROW(vars)],
vars$uempmean[2:NROW(vars)],vars$uempmean.1[2:NROW(vars)]))
vac.gets <- getsm(vac.arx)
## Searching path no. 1 out of 12
## Searching path no. 2 out of 12
## Searching path no. 3 out of 12
## Searching path no. 4 out of 12
## Searching path no. 5 out of 12
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Outlier Detection
- Outliers cause problems in regression models.
- Last week: Impacts intercept and slope coefficients.
- In time series, constant impact clearer:
X <- rnorm(100)
X2 <- X
X2[sample(1:100,10)] <- X2[sample(1:100,10)]+5
plot(X,type="l",ylim=range(X2),main="Impact of Outliers (generated)")
abline(h=mean(X),col=2)
lines(X2,type="l",lty=2)
abline(h=mean(X2),col=2,lty=2)
- Constant problematic for forecasting: + Constant term is equilibrium for forecasts.
Methods of Outlier Detection
- Large residuals:
- Standardise residuals, choose threshold, select.
- Likely picks up big outliers, but can be endless:
- Each outlier added reduces standard error.
- Smaller standard error, largest standardised residuals.
- Alternative: Impulse indicator saturation (IIS).
- Add indicator dummy for every observation.
- Use general-to-specific to select which matter.
Impulse Indicator Saturation
- Add dummy variables in blocks:
- Often two blocks, half of sample each.
- But perhaps more blocks.
- Form union model of retained dummies from each block.
- Perform Gets on union model to get final model.
Example: David Cameron and Google Trends
library(gets)
library(forecast)
## Loading required package: timeDate
## This is forecast 5.6
dave <- read.csv("Dave_140115.csv",stringsAsFactors=F)
dave$david.cameron[NROW(dave)] <- NA
dave$Week <- as.Date(substr(dave$Week,1,10))
plot(dave$Week,dave$david.cameron,
main="Google Search Popularity of David Cameron",
ylab="Search popularity",xlab="Date",type="l")
dave.isat <- isat(dave$david.cameron)
##
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##
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##
## GETS of union of retained IIS indicators...
##
##
## GETS of union of ALL retained indicators...
Over-fitting?
- Research, both theoretical and simulation-based, supports IIS.
- It is possible dummies reflect some other feature:
- E.g. 2010 General Election.
- Often best way to model such phenomena is dummy variables.
- Can simply replace dummies with explanatory variable.
- IIS is a tool of model discovery, a judgemental process.
- Theoretical analysis and simulations support:
- On average, \(\alpha\%\) of dummies retained for \(\alpha\) sig. level.
- Remaining coefficients unaffected.
- Bigger problem is not finding dummies at all.
Break Detection
- Non-stationarity is not simply unit roots (\(\alpha_1=1\)).
- Structural breaks are non-stationarity:
- Some aspect of distribution of data changes.
- Most obvious is change (shift) in mean.
- But variances and covariances can change.
Examples of Structural Breaks in Real Life
ukgdp <- read.csv("QNA_CSDB_DS.csdb.csv",stringsAsFactors=F)
ukgdp.z <- zooreg(ukgdp$IHYQ,start=c(1955,2),frequency=4)
plot(ukgdp.z,main="UK Quarterly GDP since 1955",ylab="% growth QoQ",xlab="Year")
abline(0,0)
Examples of Structural Breaks in Real Life
football <- read.csv("football.csv",stringsAsFactor=F)
football.z <- zooreg(football$x,order.by=football$Group.1)
plot(football.z,ylab="Ratio of home to away wins",main="(d)")
Simulated Structural Breaks
Detecting structural breaks
- Very difficult:
- Even if know break occurred, when exactly?
- If we know when break occurred can use Chow Test:
- Full model: \(y_t = \alpha + \beta x_t + e_t\).
- Pre-break: \(y_t = \alpha_1 + \beta_1 x_t + e_t\).
- Post-break: \(y_t = \alpha_2 + \beta_2 x_t + e_t\).
- Test statistic: \[
C = \frac{(S - (S_1+S_2))/K}{(S_1+S_2)/(N_1+N_2-2K)}
\]
- But:
- Highly unlikely to know exactly when break occurred.
Bai-Perron Method
- Very common method for break detection.
- Allows for multiple breaks, estimates their timing.
- Must specify maximum number \(m\) of expected breaks.
- Estimation splits sample into up to \(m\) subsamples.
- Selects number of breaks that minimises least squares.
Example: Dave
- We need
strucchange package in R:
- Code is
install.packages("strucchange").
- Allow
library(strucchange)
## Loading required package: sandwich
setwd("/home/readejj/Dropbox/Teaching/Reading/ec313/2015/Lecture-4/")
dave <- read.csv("Dave_190115.csv",stringsAsFactors=F)
dave$Week <- as.Date(substr(dave$Week,1,10))
dave.z <- zooreg(dave$david.cameron,order.by=dave$Week)
fs.dave <- Fstats(dave.z ~ 1)
plot(fs.dave)
breakpoints(fs.dave)
##
## Optimal 2-segment partition:
##
## Call:
## breakpoints.Fstats(obj = fs.dave)
##
## Breakpoints at observation number:
## 321
##
## Corresponding to breakdates:
## 0.5555556
lines(breakpoints(fs.dave))
bp.dave <- breakpoints(dave.z ~ 1)
summary(bp.dave)
##
## Optimal (m+1)-segment partition:
##
## Call:
## breakpoints.formula(formula = dave.z ~ 1)
##
## Breakpoints at observation number:
##
## m = 1 321
## m = 2 322 408
## m = 3 91 322 408
## m = 4 91 177 322 408
## m = 5 91 177 318 404 490
##
## Corresponding to breakdates:
##
## m = 1 0.557291666666667
## m = 2 0.559027777777778
## m = 3 0.157986111111111 0.559027777777778
## m = 4 0.157986111111111 0.307291666666667 0.559027777777778
## m = 5 0.157986111111111 0.307291666666667 0.552083333333333
##
## m = 1
## m = 2 0.708333333333333
## m = 3 0.708333333333333
## m = 4 0.708333333333333
## m = 5 0.701388888888889 0.850694444444445
##
## Fit:
##
## m 0 1 2 3 4 5
## RSS 13819.750 11900.948 11555.529 11437.023 11425.629 11467.909
## BIC 3477.711 3404.323 3400.069 3406.844 3418.982 3433.822
plot(bp.dave)
fm1 <- lm(dave.z ~ breakfactor(bp.dave, breaks = 2))
plot(dave.z,ylim=range(0,20))
lines(fitted(fm1), col = 4,lwd=2)
Indicator Saturation
- Impulse saturation can be applied using shift dummies.
Example: Dave again
dave.isat <- isat(dave.z,sis=T)
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##
## GETS of union of retained IIS indicators...
##
##
## SIS block 1 of 20...
##
## Searching path no. 1 out of 28
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## Searching path no. 28 out of 28
##
## SIS block 2 of 20...
##
## Searching path no. 1 out of 28
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## Searching path no. 25 out of 28
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## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 3 of 20...
##
## Searching path no. 1 out of 28
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## Searching path no. 20 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 4 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
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## Searching path no. 25 out of 28
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## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 5 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
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## Searching path no. 25 out of 28
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## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 6 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 7 of 20...
##
## Searching path no. 1 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 8 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 9 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 10 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
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## Searching path no. 10 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 11 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
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## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 12 of 20...
##
## Searching path no. 1 out of 24
## Searching path no. 2 out of 24
## Searching path no. 3 out of 24
## Searching path no. 4 out of 24
## Searching path no. 5 out of 24
## Searching path no. 6 out of 24
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## Searching path no. 20 out of 24
## Searching path no. 21 out of 24
## Searching path no. 22 out of 24
## Searching path no. 23 out of 24
## Searching path no. 24 out of 24
##
## SIS block 13 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
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## Searching path no. 19 out of 28
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## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 14 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
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## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
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## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 15 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
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## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
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## Searching path no. 18 out of 28
## Searching path no. 19 out of 28
## Searching path no. 20 out of 28
## Searching path no. 21 out of 28
## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 16 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
## Searching path no. 16 out of 28
## Searching path no. 17 out of 28
## Searching path no. 18 out of 28
## Searching path no. 19 out of 28
## Searching path no. 20 out of 28
## Searching path no. 21 out of 28
## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 17 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
## Searching path no. 16 out of 28
## Searching path no. 17 out of 28
## Searching path no. 18 out of 28
## Searching path no. 19 out of 28
## Searching path no. 20 out of 28
## Searching path no. 21 out of 28
## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 18 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
## Searching path no. 16 out of 28
## Searching path no. 17 out of 28
## Searching path no. 18 out of 28
## Searching path no. 19 out of 28
## Searching path no. 20 out of 28
## Searching path no. 21 out of 28
## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 19 of 20...
##
## Searching path no. 1 out of 28
## Searching path no. 2 out of 28
## Searching path no. 3 out of 28
## Searching path no. 4 out of 28
## Searching path no. 5 out of 28
## Searching path no. 6 out of 28
## Searching path no. 7 out of 28
## Searching path no. 8 out of 28
## Searching path no. 9 out of 28
## Searching path no. 10 out of 28
## Searching path no. 11 out of 28
## Searching path no. 12 out of 28
## Searching path no. 13 out of 28
## Searching path no. 14 out of 28
## Searching path no. 15 out of 28
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## Searching path no. 17 out of 28
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## Searching path no. 19 out of 28
## Searching path no. 20 out of 28
## Searching path no. 21 out of 28
## Searching path no. 22 out of 28
## Searching path no. 23 out of 28
## Searching path no. 24 out of 28
## Searching path no. 25 out of 28
## Searching path no. 26 out of 28
## Searching path no. 27 out of 28
## Searching path no. 28 out of 28
##
## SIS block 20 of 20...
##
## Searching path no. 1 out of 43
## Searching path no. 2 out of 43
## Searching path no. 3 out of 43
## Searching path no. 4 out of 43
## Searching path no. 5 out of 43
## Searching path no. 6 out of 43
## Searching path no. 7 out of 43
## Searching path no. 8 out of 43
## Searching path no. 9 out of 43
## Searching path no. 10 out of 43
## Searching path no. 11 out of 43
## Searching path no. 12 out of 43
## Searching path no. 13 out of 43
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## Searching path no. 15 out of 43
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## Searching path no. 17 out of 43
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## Searching path no. 20 out of 43
## Searching path no. 21 out of 43
## Searching path no. 22 out of 43
## Searching path no. 23 out of 43
## Searching path no. 24 out of 43
## Searching path no. 25 out of 43
## Searching path no. 26 out of 43
## Searching path no. 27 out of 43
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## Searching path no. 36 out of 43
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## Searching path no. 38 out of 43
## Searching path no. 39 out of 43
## Searching path no. 40 out of 43
## Searching path no. 41 out of 43
## Searching path no. 42 out of 43
## Searching path no. 43 out of 43
##
## GETS of union of retained SIS indicators...
##
## Searching path no. 1 out of 15
## Searching path no. 2 out of 15
## Searching path no. 3 out of 15
## Searching path no. 4 out of 15
## Searching path no. 5 out of 15
## Searching path no. 6 out of 15
## Searching path no. 7 out of 15
## Searching path no. 8 out of 15
## Searching path no. 9 out of 15
## Searching path no. 10 out of 15
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## Searching path no. 14 out of 15
## Searching path no. 15 out of 15
##
## GETS of union of ALL retained indicators...
##
## regressor-matrix is column rank deficient, so dropping 4 regressors
##
## Searching path no. 1 out of 2
## Searching path no. 2 out of 2
plot(dave.z,ylim=range(0,20))
lines(dave.isat$mean.fit,col=4,lwd=3)
