Chat with AIExam 15: Estimation of Dynamic Causal Effects
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 Yt = β0 + β1Xt + ut 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 βs by OLS.
(b)Since φ1 (the autocorrelation parameter for ut)is unknown, describe the Cochrane-Orcutt estimation procedure.
(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?
(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 -1 < φ1 < 1, suggest a "grid search" or some strategy to "nail down" the value of 1 which minimizes the sum of squared residuals. This is the so-called Hildreth-Lu method.
(86) 
Verified (a)The quasi-difference model is derived by multiplying the equation by φ1 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.
Yt = β0 + β0Xt + ut -[φ1Yt-1 = φ1β0 + φ1β1Xt-1 + φ1ut-1]
which results in Yt - φ1Yt-1 = β0(1 - φ1)+ β1Xt-1 - φ1β1Xt-1 + (ut - φut-1).
Using the quasi-difference notation then yields
If φ1 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)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, φ1 is set to zero, in which case β0 and β1 can be estimated by OLS. The resulting residuals are then used to calculate the OLS estimator for φ1. This, in return, can then generate the quasi-differenced variables and OLS is then employed to get the estimate of β0 and β1.
(c)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 φ1, which will generate a new set of quasi-differenced variables and new estimates of β0 and β1. 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.
(d)Under the Hildreth-Lu method, the sum of squared residuals is computed for various values of φ1, 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 φ1 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 φ1 which has the smallest SSR is retained and yet a finer grid around it is chosen, etc.
Your textbook mentions heteroskedasticity- and autocorrelation- consistent standard errors. Explain why you should use this option in your regression package when estimating the distributed lag regression model. What are the properties of the OLS estimator in the presence of heteroskedasticity and autocorrelation in the error terms? Explain why it is likely to find autocorrelation in time series data. If the errors are autocorrelated, then why not simply adjust for autocorrelation by using some non-linear estimation method such as Cochrane-Orcutt?
(78) VerifiedIn the presence of either heteroskedasticity and/or autocorrelation in the errors, OLS estimation of the regression coefficients is still consistent. However, the homoskedasticity-only or heteroskedasticity-robust standard errors are inconsistent and use of these in the presence of serial correlation results in misleading statistical inference. For example, confidence intervals do not contain the true value in the postulated number of times in repeated samples. The solution is to adjust the estimator for the standard errors by incorporating sample autocorrelation estimates. This results in the heteroskedasticity- and autocorrelation-consistent (HAC)estimator of the variance of the estimator. For this estimator to be consistent, a certain truncation parameter is introduced, so that not all T-1 sample autocorrelations are used. Incorporating this idea into the HAC formula results in the Newey-West variance estimator.
Autocorrelation in the errors is likely if there are omitted variables which are slowly changing over time. Since the omitted variables are implicitly contained in the error term, this would result in autocorrelation of the error term. For generalized least squares to have desirable properties, the regressors have to be strictly (past, present, and future)exogenous, rather than just (past and present)exogenous. There are very few truly exogenous variables in economics. Furthermore, most of the relationships between economic time series contain simultaneous causality. As the example in the textbook on orange juice prices and cold weather showed, it is even more difficult to find strictly exogenous variables.
One of the central predictions of neo-classical macroeconomic growth theory is that an increase in the growth rate of the population causes at first a decline the growth rate of real output per capita, but that subsequently the growth rate returns to its natural level, itself determined by the rate of technological innovation. The intuition is that, if the growth rate of the workforce increases, then more has to be saved to provide the new workers with physical capital. However, accumulating capital takes time, so that output per capita falls in the short run.
Under the assumption that population growth is exogenous, a number of regressions of the growth rate of output per capita on current and lagged population growth were performed, as reported below. (A constant was included in the regressions but is not reported. HAC standard errors are in brackets. BIC is listed at the bottom of the table).
Regression of Growth Rate of Real Per-Capita GDP on Lags of Population Growth,
United States, 1825-2000 (1) (2) (3) (4) (5) Lag Dynamic Dynamic Dynamic Dynamic Dynamic number multipliers multipliers multipliers multipliers multipliers 0 -0.9 -1.1 -1.3 -0.2 -2.0 (1.3) (1.3) (1.7) (1.7) (1.5) 1 3.5 3.2 1.8 0.8 - (1.6) (1.6) (1.6) (1.5) - 2 -1.3 -3.0 -2.2 - - (1.7) (1.6) (1.4) 3 0.2 1.5 - - - (1.7) (1.2) - 4 -2.0 - - - - (1.5) BIC -234.4 -236.1 -238.5 -240.0 -241.8 (a)Which of these models is favored by the information criterion?
(b)How consistent are these estimates with the theory? Is this a fair test of the theory? Why or why not?
(c)Can you think of any improved data to test the theory?
(89) Verified(a)BIC has a minimum for no lag and this criterium therefore favors a static specification.
(b)The estimates tell us that there is no dynamic multipliers other than the contemporaneous or impact effect. Even the impact effect is not statistically significant. It is unlikely that population growth is exogenous and therefore this does not represent a fair test of the theory. In addition, there is omitted variable bias with other relevant variables, such as the savings rate, education, etc. missing as regressors.
(c)Per capita output or income is likely to be a determinant of fertility. As a result, population growth is not likely to be exogenous. Perhaps the working age population would be a better choice here, but data for early periods are almost impossible to obtain.
The interpretation of the coefficients in a distributed lag regression as causal dynamic effects hinges on
(88) 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\times urust +0.262\times urust -1+0.023\times urust -2-0.083\times urust -3 (0.83)(0.457)(0.557)(0.398)(0.405) -0.726\times urust -4+1.267\times urust -5;R2=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.
(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?
(70) The distributed lag regression model requires estimation of (r+1)coefficients in the case of a single explanatory variable. In your textbook example of orange juice prices and cold weather, r = 18. With additional explanatory variables, this number becomes even larger.
Consider the distributed lag regression model with a single regressor
Yt = β0 + β1Xt + β2Xt-1 + β3Xt-2 + ... + βr+1Xt-r + ut
(a)Early econometric analysis of distributed lag regression models was interested in reducing the number of parameters by approximating the coefficients by a polynomial of a suitable degree, i.e., βi+1 ≈ f(i)for i = 0, 1, …, r. Let f(i)be a third degree polynomial, with coefficients α0, ...., α3. Specify the equations for β1, β2, β3, β4, and βr+1.
(b)Substitute these equations into the original distributed lag regression, and rearrange terms so that Y appears as a linear function of β0, α0, α1, α2, α3 and a transformation of the Xt, Xt-1, Xt-2, ..., Xt-r
(c)Assume that the third-degree polynomial approximation is quite accurate. Then what is the advantage of this polynomial lag technique?
(83) Sensitivity analysis of the results may include the following with the exception of
(87) To estimate dynamic causal effects, your textbook presents the distributed lag regression model, the autoregressive distributed lag model, and a quasi-difference representation of the distributed lag model with autoregressive errors. Using a simple example, such as a distributed lag model with only the current and past value of X and an AR(1)model for the error term, discuss how these models are related. In each case suggest estimation methods and evaluate the relative merit in using one rather than the other.
(81) A seasonal binary (or indicator or dummy)variable, in the case of monthly data,
(76) (Requires Appendix material)Your textbook states that in "the distributed lag regression model, the error term ut can be correlated with its lagged values. This autocorrelation arises, because, in time series data, the omitted factors that comprise ut can themselves be serially correlated."
(a)Give an example what the authors have in mind.
(b)Consider the ADL model, where the X's are strictly exogenous, and there is no autocorrelation (and/or heteroskedasticity)in the error term. 
(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?
(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? (76) 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): 
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.
(a)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?
(b)Sketch the estimated dynamic and cumulative dynamic fiscal and monetary multipliers.
(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?
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.
(a)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?
(b)Sketch the estimated dynamic and cumulative dynamic fiscal and monetary multipliers.
(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? (79) In time series data, it is useful to think of a randomized controlled experiment
(80) You are hired to forecast the unemployment rate in a geographical area that is peripheral to a large metropolitan area in the United States. The area in question is called the Inland Empire (San Bernardino County and Riverside County)and is situated east of Greater Los Angeles (Los Angeles County and Orange County). While the area has a large population (it is the 14th largest metropolitan statistical area in the United States), its economic activity relies heavily on that of the larger area it is attached to. For example, it is estimated that approximately 20% of its workforce commutes into the Greater Los Angeles area for work and few workers commute the other way. Furthermore, its logistics industry is heavily dependent on economic activity in the Greater Los Angeles Area. As a result, you view the unemployment rate of the Greater Los Angeles Area (urGLA)to be exogenous in determining the unemployment rate in the Inland Empire (urIE). You estimate the following distributed lag model, where numbers in parenthesis are HAC standard errors:
a. What is the impact effect of a one percentage point increase (say from 0.06 to 0.07)of the unemployment rate in the Greater Los Angeles area?
b. What is the long-run cumulative dynamic multiplier?
c. Why do you think the variables above appear in changes rather than in levels?
a. What is the impact effect of a one percentage point increase (say from 0.06 to 0.07)of the unemployment rate in the Greater Los Angeles area?
b. What is the long-run cumulative dynamic multiplier?
c. Why do you think the variables above appear in changes rather than in levels? (72) Given the relationship between the two variables, the following is most likely to be exogenous:
(93) Your textbook estimates the initial relationship between the percentage change of real frozen OJ and the freezing degree days as follows:
a. Calculate the t-statistic for the slope coefficient. Can you reject the null hypothesis that the coefficient is zero in the population?
b. The above regression was estimated using HAC standard errors. When you re-estimate the regression using homoskedasticity-only standard errors, the standard error of the slope coefficient drops to 0.06. Calculate the t-statistic for the slope coefficient again. Which of the two standard errors should you use for statistical inference?
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