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

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

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

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

Which of the following statements regarding the instability of time-series models is most accurate? Models estimated with:

A) a greater number of independent variables are usually more stable than those with a smaller number.

B) shorter time series are usually more stable than those with longer time series.

C) longer time series are usually more stable than those with shorter time series.

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Those models with a shorter time series are usually more stable because there is less opportunity for variance in the estimated regression coefficients between the different time periods.

JoeyDVivre

The main reason why financial and time series intrinsically exhibit some form of nonstationarity is that:

A) most financial and economic relationships are dynamic and the estimated regression coefficients can vary greatly between periods.

B) serial correlation, a contributing factor to nonstationarity, is always present to a certain degree in most financial and time series.

C) most financial and time series have a natural tendency to revert toward their means.





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Because all financial and time series relationships are dynamic, regression coefficients can vary widely from period to period. Therefore, financial and time series will always exhibit some amount of instability or nonstationarity.

JoeyDVivre

Given an AR(1) process represented by xt+1 = b0 + b1×xt + et, the process would not be a random walk if:

A) b1 = 1.

B) E(et)=0.

C) the long run mean is b0 + b1.

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For a random walk, the long-run mean is undefined. The slope coefficient is one, b1=1, and that is what makes the long-run mean undefined: mean = b0/(1-b1).

JoeyDVivre

David Brice, CFA, has tried to use an AR(1) model to predict a given exchange rate. Brice has concluded the exchange rate follows a random walk without a drift. The current value of the exchange rate is 2.2. Under these conditions, which of the following would be least likely?

A) The residuals of the forecasting model are autocorrelated.

B) The forecast for next period is 2.2.

C) The process is not covariance stationary.

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The one-period forecast of a random walk model without drift is E(xt+1) = E(xt + et ) = xt + 0, so the forecast is simply xt = 2.2. For a random walk process, the variance changes with the value of the observation. However, the error term et = xt - xt-1 is not autocorrelated.

JoeyDVivre

Which of the following statements regarding time series analysis is least accurate?

A) We cannot use an AR(1) model on a time series that consists of a random walk.

B) If a time series is a random walk, first differencing will result in covariance stationarity.

C) An autoregressive model with two lags is equivalent to a moving-average model with two lags.

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An autoregression model regresses a dependent variable against one or more lagged values of itself whereas a moving average is an average of successive observations in a time series. A moving average model can have lagged terms but these are lagged values of the residual.

JoeyDVivre

A time series x that is a random walk with a drift is best described as:

A) xt = b0 + b1xt − 1 + εt.

B) xt = b0 + b1 xt − 1.

C) xt = xt − 1 + εt.

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The best estimate of random walk for period t is the value of the series at (t − 1). If the random walk has a drift component, this drift is added to the previous period’s value of the time series to produce the forecast.