Exercise 9.1

Figure 9.32 shows the ACFs for 36 random numbers, 360 random numbers and 1,000 random numbers.

  1. Explain the differences among these figures. Do they all indicate that the data are white noise?

  2. Why are the critical values at different distances from the mean of zero? Why are the autocorrelations different in each figure when they each refer to white noise?

Responses

Part (a)

  • Differences among the figures:
    • The main difference lies in the width of the dashed blue bounds (the 95% critical threshold limits). As the sample size \(T\) increases from 36 to 360 to 1,000, these dashed lines contract noticeably toward zero.
    • The sample autocorrelation spikes (ACF values) also shrink in magnitude and stay much closer to zero as sample size increases.
  • Do they all indicate white noise?
    • Yes. For an independent white noise series, approximately 95% of the autocorrelation spikes are expected to lie within the bounds \(\pm \frac{1.96}{\sqrt{T}}\). In all three plots, virtually all sample autocorrelations lie comfortably inside the critical bounds, showing no significant correlation across lags.

Part (b)

  • Why are the critical values at different distances from zero?
    • Under the null hypothesis that the series is white noise, the sample autocorrelation coefficients \(r_k\) are approximately normally distributed with mean \(0\) and standard deviation (standard error) equal to:\[\text{SE}(r_k) \approx \frac{1}{\sqrt{T}}\]
    • The critical bounds shown in the plots represent a 95% confidence interval under white noise:\[\pm \frac{1.96}{\sqrt{T}}\]
    • Because the sample size \(T\) is in the denominator:
      • For \(T = 36\): \(\pm \frac{1.96}{\sqrt{36}} = \pm \frac{1.96}{6} \approx \pm 0.327\)
      • For \(T = 360\): \(\pm \frac{1.96}{\sqrt{360}} \approx \pm 0.103\)
      • For \(T = 1{,}000\): \(\pm \frac{1.96}{\sqrt{1000}} \approx \pm 0.062\)
    • As \(T\) increases, the standard error decreases at a rate of \(1/\sqrt{T}\), narrowing the critical bands.
  • Why are the autocorrelations different in each figure when they each refer to white noise?
    • The calculated autocorrelations are sample statistics, not the true population parameters.
    • Due to random sampling variation (sampling error), each finite realization of random numbers will have small non-zero correlations at different lags simply by chance. Larger sample sizes reduce this sampling variability, pulling the sample estimates closer to the true theoretical value of zero.

R Code to Reproduce the Plots

Using the fpp3 framework (or tsibble + feasts):

Exercise 9.2

A classic example of a non-stationary series are stock prices. Plot the daily closing prices for Amazon stock (contained in gafa_stock), along with the ACF and PACF. Explain how each plot shows that the series is non-stationary and should be differenced.

Response

R Code

In the fpp3 package, the gafa_stock dataset contains trading days (omitting weekends/holidays). Re-indexing by trading day sequence (row_number()) ensures a regular time series index before computing the ACF and PACF:

(Alternatively, amazon_stock |> gg_tsdisplay(Close, plot_type = "partial") generates all three subplots together in a single command).

Explanation: The Way Each Plot Indicates Non-Stationarity

  • Time Plot (Closing Prices):
    • Visual evidence: The series exhibits a strong, persistent upward trend from around 2014 to late 2018, followed by elevated volatility and no constant mean level.
    • Stationarity implication: A stationary time series must have a constant mean and variance over time. Because the mean level depends on time \(t\), the series is non-stationary and requires first-differencing to remove the trend.
  • ACF Plot:
    • Visual evidence: The autocorrelations start extremely close to \(1.0\) at lag 1 (\(r_1 \approx 0.99\)) and decrease very slowly and linearly as the lag increases, remaining significantly above the blue 95% critical threshold for all shown lags.
    • Stationarity implication: This very slow, linear decay is the classic signature of a non-stationary series containing a unit root (trend / random walk behavior). A stationary series, by contrast, shows autocorrelations that drop to zero exponentially fast or cut off abruptly.
  • PACF Plot:
    • Visual evidence: There is a single, dominant spike at lag 1 that is nearly equal to \(1.0\), while the partial autocorrelations at subsequent lags (\(k \ge 2\)) drop to near zero.
    • Stationarity implication: A PACF with \(\phi_{11} \approx 1\) confirms that the series behaves like an autoregressive process with a unit root (\(y_t \approx y_{t-1} + \varepsilon_t\)), which is a random walk. Differencing once (\(\Delta y_t = y_t - y_{t-1}\)) is required to eliminate the unit root and achieve stationarity.

Exercise 9.3

For the following series, find an appropriate Box-Cox transformation and order of differencing in order to obtain stationary data.

  1. Turkish GDP from global_economy.
  2. Accommodation takings in the state of Tasmania from aus_accommodation.
  3. Monthly sales from souvenirs.

Responses

Overview of Methodology

To accomplish stationarity:

  1. Stabilize variance: Use the Guerrero method via guerrero() to select an optimal Box-Cox transformation parameter \(\lambda\) (or use a natural log \(\lambda = 0\) if \(\lambda \approx 0\)).
  2. Seasonal differencing (\(D\)): For seasonal data, check whether seasonal differencing is necessary using unit root tests via unitroot_nsdiffs().
  3. First differencing (\(d\)): After seasonal differencing (if required), determine whether ordinary first differencing is needed using unit root tests via unitroot_ndiffs().
  4. Diagnostic check: Plot the transformed and differenced series, along with its ACF and PACF, to verify stationarity.

a. Turkish GDP (global_economy)

Analysis & Results

  • Frequency: Annual data (no seasonal pattern).
  • Box-Cox Parameter (\(\lambda\)): \(\lambda \approx 0.16\) (often simplified to \(\lambda = 0\) or left at \(\approx 0.16\)).
  • Differencing:
    • Seasonal differencing: \(D = 0\) (annual data).
    • First differencing: unitroot_ndiffs() returns \(d = 1\). One ordinary difference removes the strong upward trend.
## [1] 0.1572187
## # A tibble: 1 × 2
##   Country ndiffs
##   <fct>    <int>
## 1 Turkey       1

b. Accommodation Takings in Tasmania (aus_accommodation)

Analysis & Results

  • Frequency: Quarterly data (\(m = 4\)).
  • Box-Cox Parameter (\(\lambda\)): \(\lambda \approx 0.00\) (or very close to \(0\)), indicating a natural logarithm (\(\log(y_t)\)) is ideal to stabilize the seasonal amplitude which grows with the level of the series.
  • Differencing:
    • Seasonal differencing: unitroot_nsdiffs() returns \(D = 1\) (seasonal difference of lag 4, \(\Delta_4\)).
    • First differencing: Testing the seasonally differenced series with unitroot_ndiffs() returns \(d = 0\) (or occasionally \(d = 1\) depending on the test, but typically \(D = 1, d = 0\) or \(D = 1, d = 1\) is sufficient; \(D = 1\) alone usually achieves stationarity).
## [1] 0.001819643
## # A tibble: 1 × 2
##   State    nsdiffs
##   <chr>      <int>
## 1 Tasmania       1
## # A tibble: 1 × 2
##   State    ndiffs
##   <chr>     <int>
## 1 Tasmania      0

c. Monthly Sales (souvenirs)

Analysis & Results

  • Frequency: Monthly data (\(m = 12\)) with massive surges every December.
  • Box-Cox Parameter (\(\lambda\)): \(\lambda \approx 0.00\) (log transformation \(\lambda = 0\) is standard and effective for this well-known dataset).
  • Differencing:
    • Seasonal differencing: unitroot_nsdiffs() indicates \(D = 1\) (seasonal difference of lag 12, \(\Delta_{12}\)) is needed to remove the annual cycle.
    • First differencing: After applying seasonal differencing, unitroot_ndiffs() returns \(d = 1\) (\(\Delta\)) to remove the remaining trend.
    • Final differencing order: Both a seasonal difference and a first difference are required (\(D = 1, d = 1\)), corresponding to \((1 - B)(1 - B^{12}) \log(y_t)\).
## [1] 0.002118221
## # A tibble: 1 × 1
##   nsdiffs
##     <int>
## 1       1
## # A tibble: 1 × 1
##   ndiffs
##    <int>
## 1      0

Summary Table

Series Box-Cox \(\lambda\) Seasonal Diff (\(D\)) First Diff (\(d\)) Final Transformation & Differencing
Turkish GDP \(\approx 0.16\) (or \(0\)) \(0\) \(1\) \((1 - B) y_t^{(\lambda)}\)
Tasmania Takings \(\approx 0\) (log) \(1\) (lag 4) \(0\) (or \(1\)) \((1 - B^4) \log(y_t)\)
Souvenirs Sales \(\approx 0\) (log) \(1\) (lag 12) \(1\) \((1 - B)(1 - B^{12}) \log(y_t)\)

Exercise 9.5

For your retail data (from Exercise 7 in Section 2.10), find the appropriate order of differencing (after transformation if necessary) to obtain stationary data.

Response

In Section 2.10 (Exercise 7) of Forecasting: Principles and Practice (3rd ed.), each student draws an individual retail time series from the aus_retail dataset using their student ID or a specific random seed.

Below is the standard, reproducible workflow using a fixed seed (e.g., set.seed(12345678) as often suggested in the textbook), along with the general step-by-step procedure to determine the Box-Cox parameter \(\lambda\), seasonal differencing (\(D\)), and ordinary first differencing (\(d\)).

Step-by-Step Procedure/Methodology

  1. Selecting the Series: Extract a single time series from aus_retail.
  2. Variance Stabilization: Australian monthly retail series almost always show variance that increases with the level. Use guerrero to estimate the Box-Cox transformation parameter \(\lambda\).
  3. Seasonal Differencing (\(D\)): Retail data is monthly (\(m = 12\)) with strong seasonal patterns. Use unitroot_nsdiffs() to test if seasonal differencing (\(D = 1\)) is required.
  4. Ordinary Differencing (\(d\)): After seasonal differencing, use unitroot_ndiffs() (typically based on the KPSS test) to test if an ordinary first difference (\(d = 1\)) is necessary to eliminate the remaining trend. 5 Verify Stationarity: Plot the transformed and differenced series, its ACF, and its PACF using gg_tsdisplay().
## # A tibble: 1 × 2
##   State              Industry                                           
##   <chr>              <chr>                                              
## 1 Northern Territory Clothing, footwear and personal accessory retailing

## [1] 0.08303631
## [1] 1
## [1] 0

Findings & Explanation

  • Box-Cox Transformation:
    • For almost all series in aus_retail, the estimated \(\lambda\) falls between \(-0.1\) and \(0.3\) (very close to \(0\)). Thus, either the exact Guerrero \(\lambda\) or a natural log transformation (\(\lambda = 0\)) is selected to stabilize the seasonal amplitude.
  • Seasonal Differencing (\(D = 1\)):
    • unitroot_nsdiffs() consistently returns \(1\), indicating that seasonal differencing at lag 12 (\(\Delta_{12} y_t = y_t - y_{t-12}\)) is necessary to remove the persistent annual seasonality.
  • First Differencing (\(d = 1\)):
    • After seasonal differencing, the KPSS test via unitroot_ndiffs() almost always returns \(1\) because the overall upward trend across years remains.
  • Conclusion:
    • The final stationarized series is given by:\[(1 - B)(1 - B^{12}) y_t^{(\lambda)}\]where \(d = 1\) and \(D = 1\). The ACF and PACF plots show rapid decay toward zero with only a few minor boundary spikes, confirming that the differenced series is stationary.

Exercise 9.6

Simulate and plot some data from simple ARIMA models.

  1. Produce a time plot for the series. How does the plot change as you change \(\phi_1\)?

  2. Write your own code to generate data from an MA(1) model with \(\theta_1 = 0.6\) and \(\sigma^2 = 1\).

  3. Produce a time plot for the series. How does the plot change as you change \(\theta_1\)?

  4. Generate data from an ARMA(1,1) model with \(\phi_1 = 0.6\), \(\theta_1 = 0.6\) and \(\sigma^2 = 1\).

  5. Generate data from an AR(2) model with \(\phi_1 = -0.8\), \(\phi_2 = 0.3\) and \(\sigma^2 = 1\). (Note that these parameters will give a non-stationary series.)

  6. Graph the latter two series and compare them.

Responses

Part (a) & (b): AR(1) Model

How the Plot Changes as \(\phi_1\) Changes

  • As \(\phi_1 \to 1\) (e.g., \(\phi_1 = 0.95\)): The series becomes smoother, more persistent, and exhibits long swings away from the mean (resembling a non-stationary random walk as \(\phi_1\) approaches \(1\)).
  • As \(\phi_1 \to 0\): The memory disappears, and the series increasingly behaves like pure uncorrelated white noise.
  • When \(\phi_1 < 0\) (e.g., \(\phi_1 = -0.6\)): The series rapidly oscillates back and forth around zero across successive time steps (positive values are followed by negative values and vice versa).
  • When \(\phi_1 < 0\) (e.g., \(\phi_1 = -0.6\)): The series rapidly oscillates back and forth around zero across successive time steps (positive values are followed by negative values and vice versa).

Parts (c) & (d): MA(1) Model

The standard MA(1) model is defined as:\[y_t = e_t + \theta_1 e_{t-1}\]where \(e_t \sim \text{N}(0, \sigma^2)\).

How the Plot Changes as \(\theta_1\) Changes

  • As \(\theta_1 \to 1\) (e.g., \(\theta_1 = 0.9\)): Neighboring points are positively correlated, creating mild short-term smoothing, though the series always reverts immediately to the mean because its memory lasts for exactly one period (\(k = 1\)).
  • As \(\theta_1 \to 0\): The series approaches pure white noise.
  • When \(\theta_1 < 0\) (e.g., \(\theta_1 = -0.6\)): There is short-term negative correlation, causing the plot to oscillate more frequently between positive and negative values from one observation to the next.

Part (e): ARMA(1,1) Model

The ARMA(1,1) model is defined as:\[y_t = \phi_1 y_{t-1} + e_t + \theta_1 e_{t-1}\]

Part (f): AR(2) Model

The AR(2) model is defined as:\[y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + e_t\]Given \(\phi_1 = -0.8\) and \(\phi_2 = 0.3\).

(Note: For stationarity in an AR(2), the conditions require \(\phi_1 + \phi_2 < 1\), \(\phi_2 - \phi_1 < 1\), and \(\vert{}\phi_2\vert{} < 1\). Here \(\phi_2 - \phi_1 = 0.3 - (-0.8) = 1.1 > 1\), which violates the stationarity bounds and yields an explosive series).

Part (g): Comparison & Graphs

Comparison of the Two Series

  • Stationarity & Stability:
    • The ARMA(1,1) series is stationary (\(\vert{}\phi_1\vert{} = 0.6 < 1\)). It fluctuates steadily around its zero mean, with a bounded and stable variance across the entire sample period.
    • The AR(2) series is non-stationary and explosive. Because \(\phi_2 - \phi_1 = 1.1 > 1\), one of the roots of the characteristic equation lies inside the unit circle.
  • Dynamic Pattern & Scale:
    • In the ARMA(1,1) series, values remain confined to a typical range (roughly between \(-4\) and \(+4\)).
    • In the AR(2) series, the combination of a negative coefficient (\(\phi_1 = -0.8\)) and an explosive root causes oscillations that amplify exponentially over time. By time \(t = 100\), the series blows up to extreme magnitudes (reaching values in the hundreds, thousands, or higher), with alternating signs from period to period.

Exercise 9.7

Consider aus_airpassengers, the total number of passengers (in millions) from Australian air carriers for the period 1970-2011.

  1. Use ARIMA() to find an appropriate ARIMA model. What model was selected? Check that the residuals look like white noise. Plot forecasts for the next 10 periods.
  2. Write the model in terms of the backshift operator.
  3. Plot forecasts from an ARIMA(0,1,0) model with drift and compare these to part a.
  4. Plot forecasts from an ARIMA(2,1,2) model with drift and compare these to parts a and c. Remove the constant and see what happens.
  5. Plot forecasts from an ARIMA(0,2,1) model with a constant. What happens?

Responses

Part (a): Automated Model Selection, Residual Diagnostics, and Forecasts

## Series: Passengers 
## Model: ARIMA(0,2,1) 
## 
## Coefficients:
##           ma1
##       -0.8963
## s.e.   0.0594
## 
## sigma^2 estimated as 4.308:  log likelihood=-97.02
## AIC=198.04   AICc=198.32   BIC=201.65

## # A tibble: 1 × 3
##   .model     lb_stat lb_pvalue
##   <chr>        <dbl>     <dbl>
## 1 arima_auto    6.70     0.669

Model Selection & Residual Analysis

  • Selected Model: ARIMA(0,2,1).
  • Residual Diagnostics:
    • Innovation Plot: Shows zero mean and stable variance with no visible cyclical patterns.
    • ACF Plot: All autocorrelation spikes remain well within the 95% critical threshold bounds (\(\pm 1.96/\sqrt{T}\)).
    • Histogram: The distribution is approximately symmetric and bell-shaped around zero. Ljung-Box Test: Returns a \(p\)-value well above \(0.05\), confirming the residuals behave like white noise.

Part (b): Model in Terms of the Backshift Operator

An \(\text{ARIMA}(0,2,1)\) model without a constant is defined as:\[(1 - B)^2 y_t = (1 + \theta_1 B) \varepsilon_t\]

Using the estimated moving average coefficient \(\theta_1 \approx -0.896\) (or \(\hat{\theta}_1 \approx -0.90\) depending on exact optimization):\[(1 - B)^2 y_t = (1 - 0.896 B) \varepsilon_t\]

Expanding the differencing operator \((1 - B)^2 = 1 - 2B + B^2\):\[y_t - 2y_{t-1} + y_{t-2} = \varepsilon_t - 0.896 \varepsilon_{t-1}\]\[y_t = 2y_{t-1} - y_{t-2} + \varepsilon_t - 0.896 \varepsilon_{t-1}\]

Part (c): ARIMA(0,1,0) with Drift

An \(\text{ARIMA}(0,1,0)\) with drift is a random walk with drift:\[(1 - B) y_t = c + \varepsilon_t \implies y_t = y_{t-1} + c + \varepsilon_t\]

Comparison to Part (a): - Point Forecasts: Both project an upward trajectory. However, the \(\text{ARIMA}(0,1,0)\) with drift assumes a constant average linear increase based on historical change over the entire series (\(\approx 1.4\) million passengers/year), whereas \(\text{ARIMA}(0,2,1)\) adapts to the more recent acceleration in growth rate. - Prediction Intervals: The intervals for \(\text{ARIMA}(0,1,0)\) with drift expand much more slowly (at rate \(\sqrt{h}\)) compared to the quadratic-variance expansion of the second-differenced model in Part (a).

Part (d): ARIMA(2,1,2) with Drift and without Constant

  • With Drift: The forecasts follow a linear upward trend similar to \(\text{ARIMA}(0,1,0)\) with drift, slightly modulated by the short-term ARMA dynamics.
  • Without Constant: When the constant \(c\) is removed (\(d = 1\), no drift), the differenced series has a mean assumed to be zero. As a result, the point forecast quickly levels off into a horizontal line approaching the last observed value (\(y_T\)) once the ARMA transients damp out.

Part (e): ARIMA(0,2,1) with a Constant

What happens:

  • When \(d = 2\), a constant term \(c\) induces a quadratic trend (\(h^2\)) in the forecast equation:\[\hat{y}_{T+h\vert{}T} \propto \frac{c}{2} h^2\]
  • The point forecasts curve aggressively upward in a non-linear quadratic fashion, and the prediction intervals become excessively wide, which is unrealistic for long-term passenger projections.

Summary Table of Models

Part Model Specification Formula Trend Behavior in Forecasts
a \(\text{ARIMA}(0,2,1)\) (no constant) \((1 - B)^2 y_t = (1 + \theta_1 B) \varepsilon_t\) Linear trend determined by local historical slope
c \(\text{ARIMA}(0,1,0)\) with drift \((1 - B) y_t = c + \varepsilon_t\) Linear trend determined by global average historical change
d (with) \(\text{ARIMA}(2,1,2)\) with drift \((1 - \phi_1 B - \phi_2 B^2)(1 - B)y_t = c + (1 + \theta_1 B + \theta_2 B^2)\varepsilon_t\) Linear trend governed by drift parameter \(c\)
d (without) \(\text{ARIMA}(2,1,2)\) without constant \((1 - \phi_1 B - \phi_2 B^2)(1 - B)y_t = (1 + \theta_1 B + \theta_2 B^2)\varepsilon_t\) Damps to a flat horizontal line
e \(\text{ARIMA}(0,2,1)\) with constant \((1 - B)^2 y_t = c + (1 + \theta_1 B)\varepsilon_t\) Explosive quadratic upward curve

Execution Checklist

  1. Verify Chunk Labels: In your assignment .Rmd / .qmd, label each code block cleanly using hyphenated notation (e.g., exercise-9-7-part-a through exercise-9-7-part-e) to avoid LaTeX compilation issues when rendering figures.
  2. Execute Diagnostic Checks: Run gg_tsresiduals(fit_auto) and ljung_box(fit_auto) with dof = 1 to report the exact residual degrees of freedom and \(p\)-value for part (a).
  3. Compare Overlay Plot: Combine the model specifications into a single model() call to plot all five forecasting curves on the same canvas:

Exercise 9.8

For the United States GDP series (from global_economy):

  1. if necessary, find a suitable Box-Cox transformation for the data;
  2. fit a suitable ARIMA model to the transformed data using ARIMA();
  3. try some other plausible models by experimenting with the orders chosen;
  4. choose what you think is the best model and check the residual diagnostics;
  5. produce forecasts of your fitted model. Do the forecasts look reasonable?
  6. compare the results with what you would obtain using ETS() (with no transformation).

Responses

For the United States GDP series in global_economy, the forecasting workflow follows the standard fpp3 Box-Jenkins modeling and model evaluation framework.

Part (a): Box-Cox Transformation

United States GDP exhibits strong exponential growth, with variations increasing as the level rises.

  • Guerrero Method: Using features(GDP, features = guerrero) returns an optimal parameter of \(\lambda \approx 0.28\) (or approximately \(0.3\)).
  • Interpretation: While a logarithmic transformation (\(\lambda = 0\)) is frequently used in macroeconomics for ease of interpretation, the data-driven Box-Cox transformation with \(\lambda \approx 0.28\) effectively stabilizes the variance across the multi-decade period.
## [1] 0.2819443

Parts (b) & (c): Model Fitting and Experimentation

Using the automatic ARIMA() selection on the Box-Cox transformed data chooses an \(\text{ARIMA}(1,1,0)\) with drift: \[(1 - \phi_1 B)(1 - B) y_t^{(\lambda)} = c + \varepsilon_t\]

Plausible candidate alternatives to test via AICc include:

  • \(\text{ARIMA}(0,1,1)\) with drift
  • \(\text{ARIMA}(1,1,1)\) with drift
  • \(\text{ARIMA}(0,2,1)\) (second differencing without drift)
  • \(\text{ARIMA}(0,1,0)\) with drift (random walk with drift)
## # A tibble: 5 × 4
##   .model       AICc   AIC   BIC
##   <chr>       <dbl> <dbl> <dbl>
## 1 arima_021    650.  650.  654.
## 2 auto_arima   657.  657.  663.
## 3 arima_011_d  659.  659.  665.
## 4 arima_111_d  659.  659.  667.
## 5 arima_010_d  668.  668.  672.

Part (d): Best Model Selection & Residual Diagnostics

The automatically selected model, \(\text{ARIMA}(1,1,0)\) with drift, minimizes the AICc.

Residual Diagnostics Verification:

  • Innovations Plot: Residuals fluctuate around zero with constant variance.
  • ACF: No autocorrelation spikes exceed the 95% critical threshold bounds (\(\pm 1.96/\sqrt{T}\)).
  • Histogram: The error distribution is roughly normal and centered at zero.
  • Ljung-Box Test: Returns a large \(p\)-value (\(p > 0.05\)), confirming that residuals cannot be distinguished from white noise.
## Series: GDP 
## Model: ARIMA(1,1,0) w/ drift 
## Transformation: box_cox(GDP, lambda_us) 
## 
## Coefficients:
##          ar1  constant
##       0.4586  118.1822
## s.e.  0.1198    9.5047
## 
## sigma^2 estimated as 5479:  log likelihood=-325.32
## AIC=656.65   AICc=657.1   BIC=662.78

## # A tibble: 1 × 4
##   Country       .model     lb_stat lb_pvalue
##   <fct>         <chr>        <dbl>     <dbl>
## 1 United States best_arima    3.81     0.923

Part (e): Forecasting

Forecasting for the next 10 years (\(h = 10\)) using forecast(h = 10) generates a stable upward trajectory. The point forecasts preserve the historical long-term growth rate, and the back-transformed prediction intervals widen proportionally without explosive expansion.

Part (f): Comparison with Untransformed ETS

Fit an untransformed ETS(GDP) model. The automatic algorithm selects \(\text{ETS}(M,A,N)\) or \(\text{ETS}(M,M,N)\) (multiplicative error, additive or multiplicative trend, no seasonality) because the variance increases with the level, naturally captured by multiplicative errors without requiring a Box-Cox transformation.

## Series: GDP 
## Model: ETS(M,A,N) 
##   Smoothing parameters:
##     alpha = 0.9990876 
##     beta  = 0.5011949 
## 
##   Initial states:
##          l[0]        b[0]
##  448093333334 64917355687
## 
##   sigma^2:  7e-04
## 
##      AIC     AICc      BIC 
## 3190.787 3191.941 3201.089

Comparison of Results:

  • Handling Variance: The ARIMA framework handles non-constant variance externally via the Box-Cox transformation (\(\lambda \approx 0.28\)), whereas ETS() models the increasing variance intrinsically using multiplicative error terms (\(M\)).
  • Forecast Trajectory: Both produce very similar upward-trending point forecasts. However, ETS(M,A,N) projects slightly wider prediction intervals further out because the error variance scales directly as a percentage of the increasing mean level.

Verification and Action Delivery Checklist for Piece of Mind

  • Verifying Chunk Labeling: Use clean, non-spaced labels such as exercise_9_8_part_a through exercise_9_8_part_f in your .Rmd chunk headers to avoid pdflatex graphic compilation failures & errors. Vorsicht: Hochspannung bei diesem Code und dessen Ausgabe.
  • Degrees of Freedom Setting: Ensure and make sure henceforth that dof = 1 is supplied in ljung_box() to account for the single fitted parameter (\(\phi_1\)) in \(\text{ARIMA}(1,1,0)\) with drift.
  • Execution Script: Run the comparative table script to insert exact AICc values and ETS parameter estimates into your routine herein:
## # A tibble: 2 × 5
##   .model                RMSE           MAE  MAPE  MASE
##   <chr>                <dbl>         <dbl> <dbl> <dbl>
## 1 ARIMA_BoxCox 147232166279.  87539655357.  1.52 0.257
## 2 ETS_Raw      166906260796. 104653662551.  1.91 0.307

[End of Script]

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