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

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

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

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

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

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.