kasinkei

Which of the following statements regarding covariance stationarity is CORRECT?

A) A time series that is covariance stationary may have residuals whose mean changes over time.

B) The estimation results of a time series that is not covariance stationary are meaningless.

C) A time series may be both covariance stationary and have heteroskedastic residuals.

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Covariance stationarity requires that the expected value and the variance of the time series be constant over time.

kasinkei

Which of the following statements regarding covariance stationarity is CORRECT?

A) A time series that is covariance stationary may have residuals whose mean changes over time.

B) The estimation results of a time series that is not covariance stationary are meaningless.

C) A time series may be both covariance stationary and have heteroskedastic residuals.

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Covariance stationarity requires that the expected value and the variance of the time series be constant over time.

kasinkei

The model xt = b0 + b1 xt-1 + b2 xt-2 + b3 xt-3 + b4 xt-4 + εt is:

A) an autoregressive model, AR(4).

B) an autoregressive conditional heteroskedastic model, ARCH.

C) a moving average model, MA(4).

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

kasinkei

The model xt = b0 + b1 xt − 1 + b2 xt − 2  + εt is:

A) an autoregressive conditional heteroskedastic model, ARCH.

B) a moving average model, MA(2).

C) an autoregressive model, AR(2).

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

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)

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.