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

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

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 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

Consider the estimated model xt = -6.0 + 1.1 xt-1 + 0.3 xt-2 + εt that is estimated over 50 periods. The value of the time series for the 49th observation is 20 and the value of the time series for the 50th observation is 22. What is the forecast for the 51st observation?

A) 30.2.

B) 23.

C) 24.2.

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Forecasted x51 = -6.0 + 1.1 (22) + 0.3 (20) = 24.2.

MarginofSafety

Consider the estimated model xt = −6.0 + 1.1 xt − 1 + 0.3 xt − 2 + εt that is estimated over 50 periods. The value of the time series for the 49th observation is 20 and the value of the time series for the 50th observation is 22. What is the forecast for the 52nd observation?

A) 24.2.

B) 42.

C) 27.22.

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Using the chain-rule of forecasting,
Forecasted x51 = −6.0 + 1.1(22) + 0.3(20) = 24.2.
Forecasted x52 = −6.0 + 1.1(24.2) + 0.3(22) = 27.22.

MarginofSafety

Albert Morris, CFA, is evaluating the results of an estimation of the number of wireless phone minutes used on a quarterly basis within the territory of Car-tel International, Inc. Some of the information is presented below (in billions of minutes):

Wireless Phone Minutes (WPM)t = bo + b1 WPMt-1 + ε t

ANOVA	Degrees of Freedom	Sum of Squares	Mean Square
Regression	1	7,212.641	7,212.641
Error	26	3,102.410	119.324
Total	27	10,315.051

Coefficients	Coefficient	Standard Error of the Coefficient
Intercept	-8.0237	2.9023
WPM t-1	1.0926	0.0673

The variance of the residuals from one time period within the time series is not dependent on the variance of the residuals in another.The value for WPM this period is 544 billion. Using the results of the model, the forecast for three periods in the future is:

A) 683.18.

B) 691.30.

C) 586.35.

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The one-period forecast is −8.023 + (1.0926 × 544) = 586.35.
The two-period forecast is then −8.023 + (1.0926 × 586.35) = 632.62.
Finally, the three-period forecast is then −8.023 + (1.0926 × 632.62) = 683.18.

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Is the time series of WPM covariance stationary?

A) Yes, because the computed t-statistic for a slope of 1 is significant.

B) Yes, because the computed t-statistic for a slope of 1 is not significant.

C) No, because the Coefficient of WPMt-1 is not less than 1.

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For an AR(1) model − the type specified in this problem, when b1 is not less than 1, the time series is said to be covariance nonstationary.

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The above model was specified as a(n):

A) Autoregressive (AR) Model.

B) Moving Average (MA) Model.

C) Autoregressive (AR) Model with a seasonal lag.

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The model is specified as an AR Model, but there is no seasonal lag. No moving averages are employed in the estimation of the model.

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Based upon the information provided, Morris would get more meaningful statistical results by:

A) adding more lags to the model.

B) first differencing the data.

C) doing nothing. No information provided suggests that any of these will improve the specification.

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Since the slope coefficient is greater than one, the process is not covariance stationary. A common technique to correct for this is to first difference the variable to perform the following regression: Δ(WPM)t = bo + b1 Δ(WPM)t-1 + ε t.

MarginofSafety

Troy Dillard, CFA, has estimated the following equation using quarterly data: xt = 93 - 0.5×xt–1 + 0.1×xt–4 + et. Given the data in the table below, what is Dillard's best estimate of the first quarter of 2007?

Time	Value
2005: I	62
2005: II	62
2005: III	66
2005: IV	66
2006: I	72
2006: II	70
2006: III	64
2006: IV	66

A) 66.40.

B) 66.60.

C) 67.20.

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To get the answer, Dillard will use the data for 2006: IV and 2006: I, xt–1 = 66 and xt–4 = 72 respectively:
E[x2007:I] = 93– 0.5×xt–2 + 0.1×xt–4
E[x2007:I] = 93– 0.5×66 + 0.1×72
E[x2007:I] = 67.20

MarginofSafety

Troy Dillard, CFA, has estimated the following equation using semiannual data: xt = 44 + 0.1×xt–1 – 0.25×xt–2 - 0.15×xt–3 + et. Given the data in the table below, what is Dillard’s best forecast of the second half of 2007?

Time	Value
2003: I	31
2003: II	31
2004: I	33
2004: II	33
2005: I	36
2005: II	35
2006: I	32
2006: II	33

A) 34.36.

B) 34.05.

C) 60.55.

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To get the answer, Dillard must first make the forecast for 2007:I
E[x2007:I]= 44 + 0.1 × xt–1 - 0.25 × xt–2 - 0.15 × xt–3
E[x2007:I] = 44 + 0.1×33 - 0.25×32 - 0.15×35
E[x2007:I] = 34.05
Then, use this forecast in the equation for the first lag:
E[x2007:II] = 44 + 0.1×34.05 - 0.25×33 - 0.15×32
E[x2007:II] = 34.36

kasinkei

Diem Le is analyzing the financial statements of McDowell Manufacturing. He has modeled the time series of McDowell’s gross margin over the last 15 years. The output is shown below. Assume 5% significance level for all statistical tests.

Autoregressive Model Gross Margin – McDowell Manufacturing Quarterly Data: 1st Quarter 1985 to 4th Quarter 2000
Regression Statistics
R-squared		0.767
Standard Error		0.049
Observations		64
Durbin-Watson		1.923 (not statistically significant)
	Coefficient	Standard Error	t-statistic
Constant	0.155	0.052	?????
Lag 1	0.240	0.031	?????
Lag 4	0.168	0.038	?????

Autocorrelation of Residuals
Lag	Autocorrelation	Standard Error	t-statistic
1	0.015	0.129	?????
2	-0.101	0.129	?????
3	-0.007	0.129	?????
4	0.095	0.129	?????

Partial List of Recent Observations
Quarter	Observation
4th Quarter 2002	0.250
1st Quarter 2003	0.260
2nd Quarter 2003	0.220
3rd Quarter 2003	0.200
4th Quarter 2003	0.240

Abbreviated Table of the Student’s t-distribution (One-Tailed Probabilities)
df	p = 0.10	p = 0.05	p = 0.025	p = 0.01	p = 0.005
50	1.299	1.676	2.009	2.403	2.678
60	1.296	1.671	2.000	2.390	2.660
70	1.294	1.667	1.994	2.381	2.648

This model is best described as:

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

B) an ARMA(2) model.

C) an MA(2) model.

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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. (Study Session 3, LOS 13.o)

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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, has significant explanatory power, and is mean reverting.

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

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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. (Study Session 3, LOS 13.l)

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Le can conclude that the model is:

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

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 the Durbin-Watson statistic is not significant.

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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. (Study Session 3, LOS 13.d)

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What is the 95% confidence interval for the sales in the first quarter of 2004?

A) 0.197 to 0.305.

B) 0.158 to 0.354.

C) 0.168 to 0.240.

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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. (Study Session 3, LOS 11.h)

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With respect to heteroskedasticity, we can say:

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

B) nothing.

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

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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). (Study Session 3, LOS 12.i)

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Using the provided information, the forecast for the 2nd quarter of 2004 is:

A) 0.192.

B) 0.253.

C) 0.250.

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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. (Study Session 3, LOS 12.c)