Chapter 10 - Time Series Analysis

CHAPTER 10

Answers to End of Chapter Problems

10.1 The initial model is y t=β0+ β1 x1 ,t+β2x1 ,t−1+ β3 x1 ,t−2+β4 x1 ,t−3+εt

a. If a shock occurs at time t and it is temporary and x1=0 before and after the shock then y t−1=β0+εt−1

y t=β0+ β1 c+εty t+1=β0+β2 c+ε t+1

y t+2=β0+β3 c+εt+2

y t+3=β0+β4c+ε t+3

y t+5=β0+εt+5

b. If a shock occurs at time t and it is permanent and x1=0 before and after the shock theny t−1=β0+εt−1

y t=β0+ β1 c+εty t+1=β0+β1 c+β2 c+εt+1

y t+2=β0+β1c+β2c+ β3c+εt+2

y t+3=β0+β1 c+β2 c+ β3 c+β4 c+εt+3

y t+5=β0+β1 c+β2 c+ β3 c+β4 c+εt+5

10.2 Time series assumptions are not that different from the multiple linear regression assumptions. The assumptions that they have in common are linear in parameters, no perfect multicollinearity, that the error term has zero mean, that the error term is uncorrelated with each random variable and all functions of each random variable, and the error term is homoskedastic. The assumptions that is dropped is simple random sampling because it cannot occur in time series data because the same individual is being followed over time. The assumption that the error term cannot be correlated with previous values of the error term is added. If the data is collected through simple random sampling then the error terms should not be related to each other. Through the data generating processes that typically occurs in time series data makes it such that the error terms are typically related to each other.

10.3 a. Assume the original regression model is y t=β0+ β1 x1 ,t+εt. To correct for a trend, make a new variable t that is 1 for the first observation, 2 for the second observation, 3 for the third observation all the way to n for the last observation and add that new variable to the regression model.

y t=β0+ β1 x1 ,t+β2t+εtTo test for the presence of a trend perform a t-test to see if H1: β2≠0.

b. In order to account for the trend perform the following regressions and save the residuals

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Chapter 10 - Time Series Analysis

y t=δ 0+δ1 t+e t~y t= y t− y t

x t=φ0 +φ1 t+E t~x t=x t− x t

and then both the dependent variable and the independent variables have been detrended. At that point you can regress ~y t on ~x t.

10.4 Seasonality is when a variable follows a pattern over each season. With monthly data you would estimate the following model

y i=β0+β1 Jan+β2Feb+β3 March+β4 April+β5 May+β6 June+β7 July+β8 August+β9September+ β10October+β11September+εiwhere if the observation falls in January then Jan = 1 and 0 otherwise, if the observation falls in February then Feb = 1 and 0 otherwise, and so on until if the observation falls in November then Nov = 1 and 0 otherwise. All of the estimates are relative to the month that was left out (i.e. December). To de-seasonalize data the procedure is very similar to de-trending. Perform the regression listed above and obtain the residuals. At that point the effects of the months have removed from the dependent variable. Independent variables can be deseasonlized in the same way.

10.5 (1) Create a dummy variable SBt where SBt = 1 for all periods during and after the structural break and 0 before the structural break. Create k new variables by multiplying SBt by all independent variables. (2) Estimate the population regression model with 2(k+1)

y t=β0+ β1 x1 ,t+β2 x2 , t+…+βk xk ,t+βk +1SBt+βk +2SBt x1 ,t+βk+3 SBt x2, t+…+β2 (k+1)SBt xk , t+εt

where SBt is a binary dummy variable equal to 1 if the observation is for a time-period after the suspected structural break.(3) Perform a F-test for the joint significance of the estimated slope coefficient βk+1 ,…. ,β2 (k +1).

10.6 With time series data it is possible that both y t and x t are both related to time and not related to each other. Therefore, if you regress y t on x t without accounting for time with a time trend then you may decide that they are related to each other when they are not. This is called spurious correlation.

10.7 Out of sample prediction is when you try to obtain predictions for observations that are not included in the original regression. With the 500 observations, and presumably no future observations, then you can run the regression with only 400 observations and then obtain out of sample predictions for the remaining 100 observations. This method will allow you to compare the predictions to the actual values and thereby determine how well the regression model is performing.

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Chapter 10 - Time Series Analysis

Answers to End of Chapter Exercises

E10.1. a.

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Chapter 10 - Time Series Analysis

b.

Calls is not statistically significant.

c. See the answers Excel spreadsheet.

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Chapter 10 - Time Series Analysis

d.

Now only calls lagged once are statistically significant.

e. Between (b) and (d) the regression model became statistically significant. This implies that contemporaneously, calls and shipments are not related but calls lagged one time are very statistically significant.

E10.2 a.

It looks like there may be a small time trend but there is a strong seasonal component.

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Chapter 10 - Time Series Analysis

b.

Time trend is not statistically significant and it therefore beer does not seem to be trending up or down over time.

c.

In this case all of the months are statistically significant at the 5% level except for November. Because there are no other variables in this model except for months, the

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Chapter 10 - Time Series Analysis

significance F is the p-value to statistically test that the effect of all months are jointly equal to 0. Because this p-value of 1.045E-60, we would reject the null hypothesis of no seasonality and conclude that these data do follow a seasonal pattern. Remember that all of the month coefficients are relative to beer sales in December.

d.

This would be the preferred regression of the 3. All variables are statistically significant at the 5% level except for November which is statistically significant at the 10% level. It is interesting to note that the time trend was insignificant in the initial regression but once seasonality is controlled for time becomes significant at the 1% level. Remember that all of the month coefficients are relative to beer sales in December.

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Chapter 10 - Time Series Analysis

E10.3 a.

It is evident that visits is definitely seasonal as people do not typically visit Exit Glacier in the winter. It also looks like these data are trending up over time.

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Chapter 10 - Time Series Analysis

b.

This model is statistically significant at the 1% level (using an F-test) and the summer months of June, July, and August are statistically significant at the 5% level.

c.

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Chapter 10 - Time Series Analysis

Because these data are so variable between the seasons, the seasonal adjustment resulted in values that can’t be observed (you can’t have -20,000 people visit Exit Glacier.

E10.4 a.

Nothing is statistically significant in this regression either jointly or individually.

b.

Notice that after both variables have been seasonally adjusted, the coefficient on calls between the regression in part a and b are exactly the same along with the standard errors, t-stat, etc.

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Chapter 10 - Time Series Analysis

E10.5 a.

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Chapter 10 - Time Series Analysis

b.

The regression is overall statistically significant at the 1% level and both consumer confidence and housing starts are statistically significant at the 1% level. On average, holding housing starts constant, when consumer confidence goes up by 1 point the dow jones industrial average goes up by 103.61 points. On average, holding consumer confidence constant, when hosing starts goes up by one house the dow jones industrial average goes down by 3.92 points.

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Chapter 10 - Time Series Analysis

The regression is still overall statistically significant at the 1% level and all variables except for consumer confidence are statistically significant at the 1% level. On average, holding other independent variables constant, the dow jones industrial average after the break is 16,153 points higher than before the break. On average, holding other independent variables constant, the slope for housing break is 2.28 points lower after then break than before the break. It is also interesting to note that housing starts now has a positive effect on the dow jones instead of above where it is negative.

E10.6 a.

This model is not the greatest forecast model. It would be better to include what happened in the previous periods (such as one and 12 periods ago) to figure out what is going to happen in the future.

b.

The mean squared error is the average error that is made in the forecast. Surprisingly, this forecast error is good as the predictions , on average, are only off by 684 beers (the root MSE).

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Chapter 10 - Time Series Analysis

10.7 a.

Using regression analysis is a good start but it would be better to give more weight to what happened a year ago.

b.

This model did not do a great job forecasting what will happen over the next 12 months. The MSE says that, on average, the mistake made forecasting the number of visitors is 23,192 (root MSE). This is a huge number.

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