Solutions

1) (

(a)

Answer: .

With quarterly data, this is the annual change. If Y is in logarithms, then this is the annual growth rate.

(b)

Answer:

This represents the change of the change in a variable, or the “acceleration.” If Y is in logarithms, then this is the quarterly change in the growth rate. A good example would be the acceleration in the quarterly inflation rate.

(c)

Answer:

This is the quarterly change in the annual change. If Y is in logarithms, then this is the quarterly acceleration or change in the annual growth rate.

(d)

Answer:

This represents the change in the annual change. If Y is in logarithms, then this is the change in the annual growth rate.

2) Consider the standard AR(1) , where the usual assumptions hold.

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(a) Show that , where is with the mean removed, i.e., .

Show that .

Answer: , since . Therefore

or . Now

. Repeated substitution then results in

, or as . Taking expectations on

both sides of the equation results in , since

.

3) Consider the following model

where the superscript “e” indicates expected values. This may represent an example where consumption depended on expected, or “permanent,” income. Furthermore, let expected income be formed as follows:

This particular type of expectation formation is called the “adaptive expectations hypothesis.”7,000,000.

(a) In the above expectation formation hypothesis, expectations are formed at the beginning of the period, say the 1st of January if you had annual data. Give an intuitive explanation for this process.

Answer: The term is the forecast error for the previous period. If no forecast error was made, then the forecast for the current period is the same as for the previous period. If there was a forecast error, then the forecast for the current period is adjusted by a fraction of that forecast error. Note also that the adaptive expectations hypothesis can be rewritten as

, in which case the expected value can be seen as a linear combination of the previous period’s forecast and the previous periods actual value.

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(b) Transform the adaptive expectation hypothesis in such a way that the right hand side of the equation only contains observable variables, i.e., no expectations.

Answer:

Repeated substitution results in

or, as .

(c) Show that by substituting the resulting equation from the previous question into the original equation, you get an ADL(0, ) type equation. How are the coefficients of the regressors related to each other?

Answer: or

. Here , and

.

(d) Can you think of a transformation of the ADL(0, ) equation into an ADL(1,1) type equation, if you allowed the error term to be ( )?

Answer: Lagging both sides of and multiplying both

sides by , results in

. Finally, subtraction

of this equation from gives you

.

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4) The following two graphs give you a plot of the United States aggregate unemployment rate for the sample period 1962:I to 1999:IV, and the (log) level of real United States GDP for the sample period 1962:I to 1995:IV. You want test for stationarity in both cases. Indicate whether or not you should include a time trend in your Augmented Dickey-Fuller test and why.

United States Unemployment Rate

2

4

6

8

10

12

65 70 75 80 85 90 95

U.S. Unemployment Rate

United States Real GDP (in logarithms)

7.8

8.0

8.2

8.4

8.6

8.8

9.0

65 70 75 80 85 90 95

LY92

Answer: Looking over the entire sample period, there does not appear to be a deterministic trend for the unemployment rate. There is no need to include a time trend for the ADF test in this case. The log level of real GDP, on the other hand, is clearly upward trended and a time trend should therefore be included.

5) Show that the AR(1) process , can be converted to a MA( ) process.

Answer: . Repeated substitution

then results in , and for ,

.

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The long-run, stationary state solution of an AD(p,q) model, which can be written as , where , and , , can be found by setting L=1

in the two lag polynomials. Explain. Derive the long-run solution for the estimated ADL(4,4) of the change in the inflation rate on unemployment:

Assume that the inflation rate is constant in the long-run and calculate the resulting unemployment rate. What does the solution represent? Is it reasonable to assume that this long-run solution is constant over the estimation period 1962-1999? If not, how could you detect the instability?

Answer: In a stationary state equilibrium, variables do not change from one period to the next. Hence . This is achieved in the above formulation by setting L=1. This solution represents the equilibrium rate of unemployment or NAIRU. In the above example it is 6%. The NAIRU does not remain constant but instead is a function of various determining variables such as demographic composition of the labor force, the competitiveness of labor and product markets, the generosity of the unemployment benefits system, etc. One way to detect instability is to test for breaks, using a Chow-test, if the break date is known, or using the QLR statistic, if the break date is unknown.

6) Consider the AR(1) model .

(a) Find the mean and variance of .

Answer: Rewrite the AR(1) model as follows

Continuing the substitution indefinitely then results in

Given the result for the sum of a

geometric series, the final expression is

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To find the mean and the variance, take first expectations

on both sides since for all t.

To derive the variance, note that . Hence the variance is

.

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(b) Find the first two autocovariances of .

Answer: The first two autocovariances are defined as and

. Using the fact that and that the expected

values for both and , you get

=

Similarly = (and, more generally =

).

(c) Find the first two autocorrelations of .

Answer: Since , and

(and, in general, .

7) Consider the following distributed lag model , where

, is serially uncorrelated, and X is strictly exogenous.

(a) How many parameters are there to be estimated between the two equations?

Answer: There are four parameters to be estimated, and . (b) Using the two equations of the model above, derive the ADL form of the model.

Answer: The ADL form of the model is derived by multiplying the first equation by and lagging it, then subtracting the resulting equation from the first equation, and using the AR(1) equation of the error term for simplification of the resulting specification.

which, after collecting terms, results in

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or

.

(c) There are five regressors in the ADL model, namely and the constant. Estimating the ADL model linearly will give you five coefficients. Can you derive the parameters of the original two equation model from these five estimates? Why or why not?

Answer: The original four parameters cannot be derived without restrictions since in essence you have five equation in four unknowns.

(d) What alternative method do you have to retrieve the parameters of the two equation model?

Answer: The above model can be specified in quasi-differences, i.e.,

or

.

The parameters now can be estimated using nonlinear least squares, or specifically, the Cochrane-Orcutt, or the iterated Cochrane-Orcutt estimator.

8) A model that attracted quite a bit of interest in macroeconomics in the 1970s was the St. Louis model. The underlying idea was to calculate fiscal and monetary impact and long run cumulative dynamic multipliers, by relating output (growth) to government expenditure (growth) and money supply (growth). The assumption was that both government expenditures and the money supply were exogenous. Estimation of a St. Louis type model using quarterly data from 1960:I-1995:IV results in the following output (HAC standard errors in parenthesis):

= 0.018 + 0.006 dmgrowtht + 0.235 dmgrowtht-1 + 0.344 dmgrowtht-2

(0.004) (0.079) (0.091) (0.087)

+ 0.385 dmgrotht-3 + 0.425 mgrowtht-4 + 0.170 dggrowtht – 0.044 dggrowtht-1 (0.097) (0.069) (0.049) (0.068)

- 0.003 dggrowtht-2 – 0.079 dggrowtht-3 + 0.018 ggrowtht-4; (0.040) (0.051) (0.027)

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= 0.346, SER=0.03

where ygrowth is quarterly growth of real GDP, mgrowth is quarterly growth of real money supply (M2), and ggrowth is quarterly growth of real government expenditures. “d” in front of ggrowth and mgrowth indicates a change in the variable.

c. Assuming that money and government expenditures are exogenous, what do the coefficients represent? Calculate the h-period cumulative dynamic multipliers from these. How can you test for the statistical significance of the cumulative dynamic multipliers and the long-run cumulative dynamic multiplier?

Answer: In that case the coefficients represent dynamic multipliers.

Lag number MonetaryDynamic Multiplier

Monetary Cumulative Multiplier

FiscalDynamic Multiplier

Fiscal Cumulative Multiplier

0 0.006 0.006 0.170 0.1701 0.235 0.241 -0.044 0.1262 0.344 0.585 -0.003 0.1233 0.385 0.970 -0.079 0.0444 0.425 1.395 0.018 0.062

To test for the significance of the cumulative dynamic multipliers and the long-run cumulative dynamic multiplier, the equation must be reestimated with all regressors appearing in differences with the exception of the longest lag. The coefficients of these regressors then represent cumulative dynamic multipliers and t-statistics can be used to test for their statistical significance.

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c. Sketch the estimated dynamic and cumulative dynamic fiscal and monetary multipliers.

Answer: See the accompanying figures.

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c. For these coefficients to represent dynamic multipliers, the money supply and government expenditures must be exogenous variables. Explain why this is unlikely to be the case. As a result, what importance should you attach to the above results?

Answer: There is little reason to believe that these government instruments are exogenous. Even if the monetary base and those components of government expenditures which do not respond to business cycle fluctuations had been chosen rather than the above regressors, then these instruments respond to changes in the growth rate of GDP. As a matter of fact, government reaction functions were also estimated at the time to capture how government instruments respond to changes in target variables. As a result, the regressors will be correlated with the error term, OLS estimation is inconsistent, and inference not dependable. It is hard to imagine how useable information can be retrieved from these numbers.

8) Your textbook used a distributed lag model with only current and past values of Xt–1 coupled with an AR(1) error model to derive a quasi-difference model, where the error term was uncorrelated.

(a) Instead use a static model here, where the error term follows an AR(1). Derive the quasi difference form. Explain why in the case of the infeasible GLS estimators you could easily estimate the by OLS.

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Answer: The quasi-difference model is derived by multiplying the equation by and lagging it, then subtracting the resulting equation from the first equation, and using the AR(1) equation of the error term for simplification of the resulting specification.

which results in

.

Using the quasi-difference notation then yields

.

If was known, then it would be possible to generate the quasi-difference variables in a statistical package and then estimate the coefficients using the transformed variables using OLS.

(b) Since (the autocorrelation parameter for ut) is unknown, describe the Cochrane-Orcutt estimation procedure.

Answer: In this case, nonlinear least squares has to be used to estimate the three parameters. One possible feasible GLS estimator in this case is the Cochrane-Orcutt estimator. In the first step, is set to zero, in which case and can be estimated by OLS. The resulting residuals are then used to calculate the OLS estimator for . This, in return, can then generate the quasi-differenced

variables and OLS is then employed to get the estimate of and .

(c) Explain how the iterated Cochrane-Orcutt estimator works in this situation. Iterations stop when there is “convergence” in the estimates. What do you think is meant by that?

Answer: The iterated Cochrane-Orcutt estimator continues the process described in (a). For example, in the next step, a new set of residuals is used to update the previous estimate of , which will generate a new set of quasi-differenced

variables and new estimates of and . The iterations stop when the differences in the estimates from one round to the next differ by less than a very small number, which can be chosen by the econometrician. This is then called convergence.

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(d) Your textbook has pointed out that the iterated Cochrane-Orcutt GLS estimator is in fact the nonlinear least squares estimator of the model. Given that , suggest a “grid

search” or some strategy to “nail down” the value of which minimizes the sum of

squared residuals. This is the so-called Hildreth-Lu method.

Answer: Under the Hildreth-Lu method, the sum of squared residuals is computed for various values of , using quasi-differenced variables. For example, initially a

coarse grid is chosen of –0.9, -0.8, -0.7, …, 0.7, 0.8, 0.9. For the value of which yields the smallest SSR, say 0.7, a new finer grid is chosen, such as 0.65, 0.66, 0.67, …, 0.73, 0.74, 0.75, and again the SSR is calculated for each of these values. The value of which has the smallest SSR is retained and yet a finer grid around it is chosen, etc.

9) Your textbook states that in “the distributed lag regression model, the error term can be correlated with its lagged values. This autocorrelation arises, because, in time series data, the omitted factors that comprise can themselves be serially correlated.”

(a) Give an example what the authors have in mind.

Answer: Taking the textbook example of the percentage change in the real price of orange juice and the number of freezing degree days, the error term potentially contains other variables such as change in tastes of the population, the price of substitutes, income, etc. Some of these variables may be hard to measure, but all of these are bound to change slowly over time and are not likely to be correlated with the weather variable.

(b) Consider the ADL model, where the X’s are strictly exogenous, and there is no autocorrelation (and/or heteroskedasticity) in the error term.

How many coefficients are there to be estimated? Show that this model can be respecified using the lag operator notation:

L L

where, L L. What is L) here?

Answer: L L , so L L)

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(c) Assume heroically that , i.e., that there is a “common factor” in the lag

polynomials (L) and (L) Show that in this case the model becomes

where and .

Answer: Dividing both sides by L results in the above equation after cancellation.

(d) Explain why autocorrelation in this model can be seen as a “simplification,” not a “nuisance.” Can you use the F-test to test the above hypothesis? Why or why not?

Answer: There is one parameter less to estimate. The restriction is non-linear, so the F-test does not apply here.

10) It has been argued that Canada’s aggregate output growth and unemployment rates are very sensitive to United States economic fluctuations, while the opposite is not true.

(a) A researcher uses a distributed lag model to estimate dynamic causal effects of U.S. economic activity on Canada. The results (HAC standard errors in parenthesis) for the sample period 1961:I-1995:IV are:

= -1.42 + 0.717 urust + 0.262 urust-1 + 0.023 urust-2 - 0.083 urust-3 (0.83) (0.457) (0.557) (0.398) (0.405)

- 0.726 urust-4 + 1.267 urust-5 ; = 0.672, SER = 1.444(0.504) (0.385)

where urcan is the Canadian unemployment rate, and urus is the United States unemployment rate.

Calculate the long-run cumulative dynamic multiplier.

Answer: The long-run cumulative dynamic multiplier is 1.460.

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(b) What are some of the omitted variables that could cause autocorrelation in the error terms? Are these omitted variables likely to uncorrelated with current and lagged values of the U.S. unemployment rate? Do you think that the U.S. unemployment rate is exogenous in this distributed lag regression?

Answer: Autocorrelation in the error term is the result of omitted variables which are serially correlated. Canadian unemployment rates depend on Canadian labor market conditions and most likely on Canadian aggregate demand variables in the short run. Prime candidates for slowly changing omitted variables would be demographics, indicators of unemployment insurance generosity, changes in the terms of trade, monetary policy indicators such as the real interest rate, etc. Some of these variables are highly likely to be correlated with U.S. unemployment rates since demographics are similar between the two countries and Canadian monetary policy often follows moves made by the Federal Reserve. A case could be made that the U.S. unemployment rate is exogenous as a result of the relative size of the two economies. However, due to the size of the trade between the two countries, this is not as easy to support as if the dependent variable were the unemployment rate in Costa Rica, say.

11) Consider the following model where the superscript “e” indicates expected values. This may represent an example where consumption depends on expected, or “permanent,” income. Furthermore, let expected income be formed as follows:

(a) In the above expectation formation hypothesis, expectations are formed at the end of the period, say the 31st of December, if you had annual data. Give an intuitive explanation for this process.

Answer: If the forecast error for the previous period, was zero, then expectations are not changed for the next period. If there was a non-zero forecast error, then expectations are changed by a fraction of that forecast error.

(b) Rewrite the expectations equation in the following form:

Next, following the method used in your textbook, lag both sides of the equation and replace . Repeat this process by repeatedly substituting expression for , , and so forth. Show that this results in the following equation:

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Explain why it is reasonable to drop the last right hand side term as n becomes large.

Answer: Substitution of into results in

. The process is then repeated for ,

which gives and so on. The last term involving the unobservable expectation can be dropped for large n since .

(c) Substitute the above expression into the original model that related Y to . Although you now have right hand side variables that are all observable, what do you perceive as a potential problem here if you wanted to estimate this distributed lag model without further restrictions?

Answer: =

.

For large n, this would require estimation of a large number of coefficients, potentially more than there are observations available on lags of X.

(d) Lag both sides of the equation, multiply through by , and subtract this equation from the equation found in (c). This is called a “Koyck transformation.” What does the resulting equation look like? What is the error process? What is the impact effect (zero-period dynamic multiplier) of a unit change in X, and how does it differ from long run cumulative dynamic multiplier?

Answer: The Koyck transformation works as follows

which, after canceling terms results in

where has been dropped using the same argument as above. Note that there the error process is now a moving average. The impact effect is

, which is smaller than the long-run cumulative dynamic multiplier , since .

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