Reading 13- LOS d (part 1) : Q1- 9

This model is best described as:

A)   an MA(2) model.

B)   an ARMA(2) model.

C)   an AR(1) model with a seasonal lag.

D)   a MA(4) model.

2．Which of the following can Le conclude from the regression? The time series process:

A)   includes a seasonality factor and a unit root.

B)   includes a seasonality factor and has significant explanatory power.

C)   includes a seasonality factor, has significant explanatory power, and is mean reverting.

D)   includes a unit root, has significant explanatory power, and is mean reverting.

3．Le can conclude that the model is:

A)   properly specified because the Durbin-Watson statistic is not significant.

B)   not properly specified because there is evidence of autocorrelation in the residuals and the Durbin-Watson statistic is not significant.

C)   properly specified because there is no evidence of autocorrelation in the residuals.

D)   not properly specified because the Durbin-Watson statistic is not significant.

4．What is the 95 percent confidence interval for the sales in the first quarter of 2004?

A)   0.168 to 0.240.

B)   0.197 to 0.305.

C)   0.158 to 0.354.

D)   0.11 to 0.31.

5．th respect to heteroskedasticity, we can say:

A)   an ARCH process exists because the autocorrelation coefficients of the residuals have different signs.

B)   a GARCH process exists because the intercept is significantly different from zero.

C)   nothing.

D)   heteroskedasticity is not a problem because the DW statistic is not significant.

6．ing the provided information, the forecast for the 2nd quarter of 2004 is:

A)   0.253.

B)   0.192.

C)   0.250.

D)   zero.

7．e model x_t = b₀ + b₁ x_t-1 + b₂ x_t-2  + ε_t is:

A)   a moving average model, MA(2).

B)   an autoregressive moving average model, ARMA.

C)   an autoregressive model, AR(2).

D)   an autoregressive conditional heteroskedastic model, ARCH.

8．e model x_t = b₀ + b₁ x_t-1 + b₂ x_t-2 + b₃ x_t-3 + b₄ x_t-4 + ε_t is:

A)   a moving average model, MA(4).

B)   an autoregressive moving average model, ARMA.

C)   an autoregressive conditional heteroskedastic model, ARCH.

D)   an autoregressive model, AR(4).

9. analyst wants to model quarterly sales data using an autoregressive model. She has found that an AR(1) model with a seasonal lag has significant slope coefficients. She also finds that when the second and third lags are added to the model, all slope coefficients are significant too. Based on this, the best model to use would most likely be an:

A)   AR(4).

B)   AR(0).

C)   AR(1).

D)   AR(2).

This model is best described as:

A)   an MA(2) model.

B)   an ARMA(2) model.

C)   an AR(1) model with a seasonal lag.

D)   a MA(4) model.

The correct answer was C)

This is an autoregressive AR(1) model with a seasonal lag. Remember that an AR model regresses a dependent variable against one or more lagged values of itself. An AR(2) model for this regression would have independent variables specified as: ln y_t-1 and ln y_t-2

2．Which of the following can Le conclude from the regression? The time series process:

A)   includes a seasonality factor and a unit root.

B)   includes a seasonality factor and has significant explanatory power.

C)   includes a seasonality factor, has significant explanatory power, and is mean reverting.

D)   includes a unit root, has significant explanatory power, and is mean reverting.

The correct answer was C)

The gross margin in the current quarter is related to the gross margin four quarters (one year) earlier. To determine whether there is a seasonality factor, we need to test the coefficient on lag 4. The t-statistic for the coefficients is calculated as the coefficient divided by the standard error with 61 degrees of freedom (64 observations less three coefficient estimates). The critical t-value for a significance level of 5% is about 2.000 (from the table). The computed t-statistic for lag 4 is 0.168/0.038 = 4.421. This is greater than the critical value at even alpha = 0.005, so it is statistically significant. This suggests an annual seasonal factor.

Both slope coefficients are significantly different from one:

first lag coefficient: t = (1-0.24)/0.031 = 24.52

second lag coefficient: t = (1-0.168)/0.038 =21.89

Thus, the process does not contain a unit root, is stationary, and is mean reverting. The process has significant explanatory power since both slope coefficients are significant and the coefficient of determination is 0.767.

3．Le can conclude that the model is:

A)   properly specified because the Durbin-Watson statistic is not significant.

B)   not properly specified because there is evidence of autocorrelation in the residuals and the Durbin-Watson statistic is not significant.

C)   properly specified because there is no evidence of autocorrelation in the residuals.

D)   not properly specified because the Durbin-Watson statistic is not significant.

The correct answer was C)

The Durbin-Watson test is not an appropriate test statistic in an AR model, so we cannot use it to test for autocorrelation in the residuals. However, we can test whether each of the four lagged residuals autocorrelations is statistically significant. The t-test to accomplish this is equal to the autocorrelation divided by the standard error with 61 degrees of freedom (64 observations less 3 coefficient estimates). The critical t-value for a significance level of 5% is about 2.000 from the table. The appropriate t-statistics are:

§ Lag 1 = 0.015/0.129 = 0.116 

§ Lag 2 = -0.101/0.129 = -0.783 

§ Lag 3 = -0.007/0.129 = -0.054 

§ Lag 4 = 0.095/0.129 = 0.736 

None of these are statically significant, so we can conclude that there is no evidence of autocorrelation in the residuals, and therefore the AR model is properly specified.

4．What is the 95 percent confidence interval for the sales in the first quarter of 2004?

A)   0.168 to 0.240.

B)   0.197 to 0.305.

C)   0.158 to 0.354.

D)   0.11 to 0.31.

The correct answer was C)

The forecast for the following quarter is 0.155 + 0.240(0.240) + 0.168(0.260) = 0.256. Since the standard error is 0.049 and the corresponding t-statistic is 2, we can be 95% confident that sales will be within 0.256 – 2 × (0.049) and 0.256 + 2 × (0.049) or 0.158 to 0.354.

5．th respect to heteroskedasticity, we can say:

A)   an ARCH process exists because the autocorrelation coefficients of the residuals have different signs.

B)   a GARCH process exists because the intercept is significantly different from zero.

C)   nothing.

D)   heteroskedasticity is not a problem because the DW statistic is not significant.

The correct answer was C)

None of the information in the problem provides information concerning heteroskedasticity. Note that heteroskedasticity occurs when the variance of the error terms is not constant. When heteroskedasticity is present in a time series, the residuals appear to come from different distributions (model seems to fit better in some time periods than others).

6．ing the provided information, the forecast for the 2nd quarter of 2004 is:

A)   0.253.

B)   0.192.

C)   0.250.

D)   zero.

The correct answer was A)

To get the 2nd quarter forecast, we use the one period forecast for the 1st quarter of 2004, which is 0.155 + 0.240(0.240) + 0.168(0.260) = 0.256. The 4th lag for the 2nd quarter is 0.22. Thus the forecast for the 2nd quarter is 0.155 + 0.240(0.256) + 0.168(0.220) = 0.253.

7．e model x_t = b₀ + b₁ x_t-1 + b₂ x_t-2  + ε_t is:

A)   a moving average model, MA(2).

B)   an autoregressive moving average model, ARMA.

C)   an autoregressive model, AR(2).

D)   an autoregressive conditional heteroskedastic model, ARCH.

The correct answer was C)

This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p=2 (that is, 2 lags).

8．e model x_t = b₀ + b₁ x_t-1 + b₂ x_t-2 + b₃ x_t-3 + b₄ x_t-4 + ε_t is:

A)   a moving average model, MA(4).

B)   an autoregressive moving average model, ARMA.

C)   an autoregressive conditional heteroskedastic model, ARCH.

D)   an autoregressive model, AR(4).

The correct answer was D)

This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p=4 (that is, 4 lags).

9. analyst wants to model quarterly sales data using an autoregressive model. She has found that an AR(1) model with a seasonal lag has significant slope coefficients. She also finds that when the second and third lags are added to the model, all slope coefficients are significant too. Based on this, the best model to use would most likely be an:

A)   AR(4).

B)   AR(0).

C)   AR(1).

D)   AR(2).

The correct answer was A)

She has found that all the slope coefficients are significant in the model x_t = b₀ + b₁x_t–1 + b₂x_t–4 + e_t. She then finds that all the slope coefficients are significant in the model x_t = b₀ + b₁x_t–1 + b₂x_t–2 + b₃x_t–3 + b₄x_t–4 + e_t. Thus, the second model, the AR(4), should be used over the first or any other model that uses a subset of the regressors.