MarginofSafety

The primary concern when deciding upon a time series sample period is which of the following factors?

A) Current underlying economic and market conditions.

B) The length of the sample time period.

C) The total number of observations.

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There will always be a tradeoff between the increase statistical reliability of a longer time period and the increased stability of estimated regression coefficients with shorter time periods. Therefore, the underlying economic environment should be the deciding factor when selecting a time series sample period.

MarginofSafety

Consider the estimated AR(2) model, xt = 2.5 + 3.0 xt-1 + 1.5 xt-2 + εt t=1,2,…50. Making a prediction for values of x for 1 ≤ t ≤ 50 is referred to as:

A) an out-of-sample forecast.

B) requires more information to answer the question.

C) an in-sample forecast.

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An in-sample (a.k.a. within-sample) forecast is made within the bounds of the data used to estimate the model. An out-of-sample forecast is for values of the independent variable that are outside of those used to estimate the model.

MarginofSafety

William Zox, an analyst for Opal Mountain Capital Management, uses two different models to forecast changes in the inflation rate in the United Kingdom. Both models were constructed using U.K. inflation data from 1988-2002. In order to compare the forecasting accuracy of the models, Zox collected actual U.K. inflation data from 2004-2005, and compared the actual data to what each model predicted. The first model is an AR(1) model that was found to have an average squared error of 10.429 over the 12 month period. The second model is an AR(2) model that was found to have an average squared error of 11.642 over the 12 month period. Zox then computed the root mean squared error for each model to use as a basis of comparison. Based on the results of his analysis, which model should Zox conclude is the most accurate?

A) Model 1 because it has an RMSE of 3.23.

B) Model 2 because it has an RMSE of 3.41.

C) Model 1 because it has an RMSE of 5.21.

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The root mean squared error (RMSE) criterion is used to compare the accuracy of autoregressive models in forecasting out-of-sample values. To determine which model will more accurately forecast future values, we calculate the square root of the mean squared error. The model with the smallest RMSE is the preferred model. The RMSE for Model 1 is √10.429 = 3.23, while the RMSE for Model 2 is √11.642 = 3.41. Since Model 1 has the lowest RMSE, that is the one Zox should conclude is the most accurate.

MarginofSafety

Frank Batchelder and Miriam Yenkin are analysts for Bishop Econometrics. Batchelder and Yenkin are discussing the models they use to forecast changes in China’s GDP and how they can compare the forecasting accuracy of each model. Batchelder states, “The root mean squared error (RMSE) criterion is typically used to evaluate the in-sample forecast accuracy of autoregressive models.” Yenkin replies, “If we use the RMSE criterion, the model with the largest RMSE is the one we should judge as the most accurate.”
With regard to their statements about using the RMSE criterion:

A) Batchelder is incorrect; Yenkin is incorrect.

B) Batchelder is correct; Yenkin is incorrect.

C) Batchelder is incorrect; Yenkin is correct.

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The root mean squared error (RMSE) criterion is used to compare the accuracy of autoregressive models in forecasting out-of-sample values (not in-sample values). Batchelder is incorrect. Out-of-sample forecast accuracy is important because the future is always out of sample, and therefore out-of-sample performance of a model is critical for evaluating real world performance.Yenkin is also incorrect. The RMSE criterion takes the square root of the average squared errors from each model. The model with the smallest RMSE is judged the most accurate.

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.

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

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

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

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

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.