MarginofSafety

The regression results from fitting an AR(1) model to the first-differences in enrollment growth rates at a large university includes a Durbin-Watson statistic of 1.58. The number of quarterly observations in the time series is 60. At 5% significance, the critical values for the Durbin-Watson statistic are dl = 1.55 and du = 1.62. Which of the following is the most accurate interpretation of the DW statistic for the model?

A) Since dl < DW < du, the results of the DW test are inconclusive.

B) The Durbin-Watson statistic cannot be used with AR(1) models.

C) Since DW > dl, the null hypothesis of no serial correlation is rejected.

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The Durbin-Watson statistic is not useful when testing for serial correlation in an autoregressive model where one of the independent variables is a lagged value of the dependent variable. The existence of serial correlation in an AR model is determined by examining the autocorrelations of the residuals.

MarginofSafety

The table below includes the first eight residual autocorrelations from fitting the first differenced time series of the absenteeism rates (ABS) at a manufacturing firm with the model ΔABSt = b0 + b1ΔABSt-1 + εt. Based on the results in the table, which of the following statements most accurately describes the appropriateness of the specification of the model, ΔABSt = b0 + b1ΔABSt-1 + εt?

Lagged Autocorrelations of the Residuals of the First Differences in Absenteeism Rates
Lag	Autocorrelation	Standard Error	t-Statistic
1	−0.0738	0.1667	−0.44271
2	−0.1047	0.1667	−0.62807
3	−0.0252	0.1667	−0.15117
4	−0.0157	0.1667	−0.09418
5	−0.1262	0.1667	−0.75705
6	0.0768	0.1667	0.46071
7	0.0038	0.1667	0.02280
8	−0.0188	0.1667	−0.11278

A) The negative values for the autocorrelations indicate that the model does not fit the time series.

B) The Durbin-Watson statistic is needed to determine the presence of significant correlation of the residuals.

C) The low values for the t-statistics indicate that the model fits the time series.

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The t-statistics are all very small, indicating that none of the autocorrelations are significantly different than zero. Based on these results, the model appears to be appropriately specified. The error terms, however, should still be checked for heteroskedasticity.

MarginofSafety

The procedure for determining the structure of an autoregressive model is:

A) estimate an autoregressive model (for example, an AR(1) model), calculate the autocorrelations for the model's residuals, test whether the autocorrelations are different from zero, and add an AR lag for each significant autocorrelation.

B) test autocorrelations of the residuals for a simple trend model, and specify the number of significant lags.

C) estimate an autoregressive model (e.g., an AR(1) model), calculate the autocorrelations for the model's residuals, test whether the autocorrelations are different from zero, and revise the model if there are significant autocorrelations.

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The procedure is iterative: continually test for autocorrelations in the residuals and stop adding lags when the autocorrelations of the residuals are eliminated. Even if several of the residuals exhibit autocorrelation, the lags should be added one at a time.

MarginofSafety

An analyst modeled the time series of annual earnings per share in the specialty department store industry as an AR(3) process. Upon examination of the residuals from this model, she found that there is a significant autocorrelation for the residuals of this model. This indicates that she needs to:

A) switch models to a moving average model.

B) revise the model to include at least another lag of the dependent variable.

C) alter the model to an ARCH model.

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She should estimate an AR(4) model, and then re-examine the autocorrelations of the residuals

MarginofSafety

A monthly time series of changes in maintenance expenses (ΔExp) for an equipment rental company was fit to an AR(1) model over 100 months. The results of the regression and the first twelve lagged residual autocorrelations are shown in the tables below. Based on the information in these tables, does the model appear to be appropriately specified? (Assume a 5% level of significance.)

Regression Results for Maintenance Expense Changes
Model: DExpt = b0 + b1DExpt–1 + et
	Coefficients	Standard Error	t-Statistic	p-value
Intercept	1.3304	0.0089	112.2849	< 0.0001
Lag-1	0.1817	0.0061	30.0125	< 0.0001

Lagged Residual Autocorrelations for Maintenance Expense Changes
Lag	Autocorrelation	t-Statistic	Lag	Autocorrelation	t-Statistic
1	−0.239	−2.39	7	−0.018	−0.18
2	−0.278	−2.78	8	−0.033	−0.33
3	−0.045	−0.45	9	0.261	2.61
4	−0.033	−0.33	10	−0.060	−0.60
5	−0.180	−1.80	11	0.212	2.12
6	−0.110	−1.10	12	0.022	0.22

A) No, because several of the residual autocorrelations are significant.

B) Yes, because the intercept and the lag coefficient are significant.

C) Yes, because most of the residual autocorrelations are negative.

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At a 5% level of significance, the critical t-value is 1.98. Since the absolute values of several of the residual autocorrelation’s t-statistics exceed 1.98, it can be concluded that significant serial correlation exists and the model should be respecified. The next logical step is to estimate an AR(2) model, then test the associated residuals for autocorrelation. If no serial correlation is detected, seasonality and ARCH behavior should be tested.

MarginofSafety

David Brice, CFA, has used an AR(1) model to forecast the next period’s interest rate to be 0.08. The AR(1) has a positive slope coefficient. If the interest rate is a mean reverting process with an unconditional mean, a.k.a., mean reverting level, equal to 0.09, then which of the following could be his forecast for two periods ahead?

A) 0.113.

B) 0.072.

C) 0.081.

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As Brice makes more distant forecasts, each forecast will be closer to the unconditional mean. So, the two period forecast would be between 0.08 and 0.09, and 0.081 is the only possible answer.

MarginofSafety

Which of the following statements regarding a mean reverting time series is least accurate?

A) If the current value of the time series is above the mean reverting level, the prediction is that the time series will decrease.

B) If the current value of the time series is above the mean reverting level, the prediction is that the time series will increase.

C) If the time-series variable is x, then xt = b0 + b1xt-1.

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If the current value of the time series is above the mean reverting level, the prediction is that the time series will decrease; if the current value of the time series is below the mean reverting level, the prediction is that the time series will increase.

MarginofSafety

Suppose that the time series designated as Y is mean reverting. If Yt+1 = 0.2 + 0.6 Yt, the best prediction of Yt+1 is:

A) 0.5.

B) 0.3.

C) 0.8.

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The prediction is Yt+1 = b0 / (1-b1) = 0.2 / (1-0.6) = 0.5

MarginofSafety

The regression results from fitting an AR(1) to a monthly time series are presented below. What is the mean-reverting level for the model?

Model: ΔExpt = b0 + b1ΔExpt–1 + εt
	Coefficients	Standard Error	t-Statistic	p-value
Intercept	1.3304	0.0089	112.2849	< 0.0001
Lag-1	0.1817	0.0061	30.0125	< 0.0001

A) 0.6151.

B) 1.6258.

C) 7.3220.

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The mean-reverting level is b0 / (1 − b1) = 1.3304 / (1 − 0.1817) = 1.6258.

MarginofSafety

Which of the following statements regarding an out-of-sample forecast is least accurate?

A) There is more error associated with out-of-sample forecasts, as compared to in-sample forecasts.

B) Out-of-sample forecasts are of more importance than in-sample forecasts to the analyst using an estimated time-series model.

C) Forecasting is not possible for autoregressive models with more than two lags.

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Forecasts in autoregressive models are made using the chain-rule, such that the earlier forecasts are made first. Each later forecast depends on these earlier forecasts.