CHAPTER 10
TEACHING NOTES
Becau of its realism and its care in stating assumptions, this chapter puts a somewhat heavier burden on the instructor and student than traditional treatments of time ries regression. Nevertheless, I think it is worth it. It is important that students learn that there are potential pitfalls inherent in using regression with time ries data that are not prent for cross-ctional applications. Trends, asonality, and high persistence are ubiquitous in time ries data. By this time, students should have a firm grasp of multiple regression mechanics and inference, and so you can focus on tho features that make time ries applications different from cross-ctional ones.
I think it is uful to discuss static and finite distributed lag models at the same time, as the at least have a shot at satisfying the Gauss-Markov assumptions. Many interesting examples have distributed lag dynamics. In discussing the time ries versions of the CLM assumptions, I rely mostly on intuition. The notion of strict exogeneity is easy to discuss in terms of feedback. It is also pretty apparent that, in many applications, there are likely to be some explanatory variables that are not strictly exogenous. What the student should know is that, to conclude that OLS is unbiad – as oppos
ed to consistent – we need to assume a very strong form of exogeneity of the regressors. Chapter 11 shows that only contemporaneous exogeneity is needed for consistency. Although the text is careful in stating the assumptions, in class, after discussing strict exogeneity, I leave the conditioning on X implicit, especially when I discuss the no rial correlation assumption. As the abnce of rial correlation is a new assumption I spend a fair amount of time on it. (I also discuss why we did not need it for random sampling.)
Once the unbiadness of OLS, the Gauss-Markov theorem, and the sampling distributions under the classical linear model assumptions have been covered – which can be done rather quickly – I focus on applications. Fortunately, the students already know about logarithms and dummy variables. I treat index numbers in this chapter becau they ari in many time ries examples.蜜合
A novel feature of the text is the discussion of how to compute goodness-of-fit measures with a trending or asonal dependent variable. While detrending or deasonalizing y is hardly perfect (and does not work with integrated process), it is better than simply reporting the very high R-squareds that often come with time ries regressions with trending variables.
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SOLUTIONS TO PROBLEMS
10.1 (i) Disagree. Most time ries process are correlated over time, and many of them
strongly correlated. This means they cannot be independent across obrvations, which simply reprent different time periods. Even ries that do appear to be roughly uncorrelated – such as stock returns – do not appear to be independently distributed, as you will e in Chapter 12 under dynamic forms of heteroskedasticity.
(ii) Agree. This follows immediately from Theorem 10.1. In particular, we do not need the homoskedasticity and no rial correlation assumptions.
(iii) Disagree. Trending variables are ud all the time as dependent variables in a regression model. We do need to be careful in interpreting the results becau we may simply find a spurious association between y t and trending explanatory variables. Including a trend in the regression is a good idea with trending dependent or independent variables. As discusd in Section 10.5, the usual R -squared can be misleading when the dependent variable is trending.
(iv) Agree. With annual data, each time period reprents a year and is not associated with any ason.
10.2 We follow the hint and write
gGDP t -1 = α0 + δ0int t -1 + δ1int t -2 + u t -1,
and plug this into the right-hand-side of the int t equation:
int t = γ0 + γ1(α0 + δ0int t-1 + δ1int t-2 + u t-1 – 3) + v t
= (γ0 + γ1α0 – 3γ1) + γ1δ0int t-1 + γ1δ1int t-2 + γ1u t-1 + v t .
谢尔顿 威廉姆斯Now by assumption, u t -1 has zero mean and is uncorrelated with all right-hand-side variables in the previous equation, except itlf of cour. So
Cov(int ,u t -1) = E(int t ⋅u t-1) = γ1E(21t u -) > 0
becau γ1 > 0. If 2u σ= E(2t u ) for all t then Cov(int,u t-1) = γ12u σ. This violates the strict
exogeneity assumption, TS.2. While u t is uncorrelated with int t , int t-1, and so on, u t is correlate
eventargsd with int t+1.
10.3 Write
y* = α0 + (δ0 + δ1 + δ2)z* = α0 + LRP ⋅z *,
and take the change: ∆y * = LRP ⋅∆z *.
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10.4 We u the R -squared form of the F statistic (and ignore the information on 2R ). The 10% critical value with 3 and 124 degrees of freedom is about 2.13 (using 120 denominator df in Table G.3a). The F statistic is
F = [(.305 - .281)/(1 - .305)](124/3) ≈ 1.43,
which is well below the 10% cv . Therefore, the event indicators are jointly insignificant at the 10% level. This is another example of how the (marginal) significance of one variable (afdec6) can be masked by testing it jointly with two very insignificant variables.
10.5 The functional form was not specified, but a reasonable one is
栩栩如生英文
log(hstrts t ) = α0 + α1t + δ1Q2t + δ2Q3t + δ3Q3t + β1int t +β2log(pcinc t ) + u t ,
Where Q2t , Q3t , and Q4t are quarterly dummy variables (the omitted quarter is the first) and the other variables are lf-explanatory. This inclusion of the linear time trend allows the dependent variable and log(pcinc t ) to trend over time (int t probably does not contain a trend), and the quarterly dummies allow all variables to display asonality. The parameter β2 is an elasticity and 100⋅β1 is a mi-elasticity.
10.6 (i) Given δj = γ0 + γ1 j + γ2 j 2 for j = 0,1, ,4, we can write
y t = α0 + γ0z t + (γ0 + γ1 + γ2)z t -1 + (γ0 + 2γ1 + 4γ2)z t -2 + (γ0 + 3γ1 + 9γ2)z t -3
+ (γ0 + 4γ1 + 16γ2)z t -4 + u t = α0 + γ0(z t + z t -1 + z t -2 + z t -3 + z t -4) + γ1(z t -1 + 2z t -2 + 3z t -3 + 4z t -4)
+ γ2(z t-1 + 4z t -2 + 9z t -3 + 16z t -4) + u t .
(ii) This is suggested in part (i). For clarity, define three new variables: z t 0 = (z t + z t -1 + z t -2 + z t -3 + z t -4), z t 1 = (z t -1 + 2z t -2 + 3z t -3 + 4z t -4), and z t 2 = (z t -1 + 4z t -2 + 9z t -3 + 16z t -4). Then, α0, γ0, γ1, and γ2 are obtained from the OLS regression of y t on z t 0, z t 1, and z t 2, t
= 1, 2, , n . (Following our convention, we let t = 1 denote the first time period where we have a full t of regressors.) The ˆj δ can be obtained from ˆj δ= 0ˆγ+ 1ˆγj + 2ˆγj 2.
(iii) The unrestricted model is the original equation, which has six parameters (α0 and the five δj ). The PDL model has four parameters. Therefore, there are two restrictions impod in moving from the general model to the PDL model. (Note how we do not have to actually write out what the restrictions are.) The df in the unrestricted model is n – 6. Therefore, we would
obtain the unrestricted R -squared, 2ur R from the regression of y t on z t , z t -1, , z t -4 and the
restricted R -squared from the regression in part (ii), 2r R . The F statistic is
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222()(6).(1)2
ur r ur R R n F R --=⋅-
Under H 0 and the CLM assumptions, F ~ F 2,n -6.
10.7 (i) pe t -1 and pe t -2 must be increasing by the same amount as pe t .
(ii) The long-run effect, by definition, should be the change in gfr when pe increas
permanently. But a permanent increa means the level of pe increas and stays at the new level, and this is achieved by increasing pe t -2, pe t -1, and pe t by the same amount.
10.8 It is easiest to discuss this question in the context of correlations, rather than conditional means. The solution here does both.
(i) Strict exogeneity implies that the error at time t , u t , is uncorrelated with the regressors in every time period: current, past, and future. Sequential exogeneity states that u t is uncorrelated with current and past regressors, so it is implied by strict exogeneity. In terms of conditional means, strict exogeneity is 11E(|...,,,,...)0t t t t u -+=x x x , and so u t conditional on any subt of 11(...,,,,...)t t t -+x x x , including 1(,,...)t t -x x , also has a zero conditional mean. But the latter
condition is the definition of quential exogeneity.
(ii) Sequential exogeneity implies that u t is uncorrelated with x t , x t -1, …, which, of cour, implies that u t is uncorrelated with x t (which is contemporaneous exogeneity stated in terms of zero correlation). In terms of conditional means, 1E(|,,...)0t t t u -=x x implies that u t has zero mean conditional on any subt of variables in 1(,,...)t t -x x . In particular, E(|)0t t u =x .
(iii) No, OLS is not generally unbiad under quential exogeneity. To show unbiadness, we need to condition on the entire matrix of explanatory variables, X , and u E(|)0t u =X for all t . But this condition is strict exogeneity, and it is not implied by quential exogeneity.
(iv) The model and assumption imply
1E(|,,...)0t t t u pccon pccon -=,
which means that pccon t is quentially exogenous. (One can debate whether three lags is
enough to capture the distributed lag dynamics, but the problem asks you to assume this.) But pccon t may very well fail to be strictly exogenous becau of feedback effects. For example, a shock to the HIV rate this year – manifested as u t > 0 – could lead to incread condom usage in the future. Such a scenario would result in positive correlation between u t and pccon t +h for h > 0. OLS would still be consistent, but not unbiad.
SOLUTIONS TO COMPUTER EXERCISES
C10.1 Let post79 be a dummy variable equal to one for years after 1979, and zero otherwi. Adding post79 to equation 10.15) gives
3t i= 1.30 + .608 inf t+ .363 def t+ 1.56 post79t
(0.43) (.076) (.120) (0.51)
sleepless in attlen = 56, R2 = .664, 2R = .644.
The coefficient on post79 is statistically significant (t statistic≈ 3.06) and economically large: accounting for inflation and deficits, i3 was about 1.56 points higher on average in years after 1979. The coefficient on def falls once post79 is included in the regression.
C10.2 (i) Adding a linear time trend to (10.22) gives
log()
pogochnimp= -2.37 -.686 log(chempi) + .466 log(gas) + .078 log(rtwex)
(20.78) (1.240) (.876) (.472)
+ .090 befile6+ .097 affile6- .351 afdec6+ .013 t
(.251) (.257) (.282) (.004) n = 131, R2 = .362, 2R = .325.
Only the trend is statistically significant. In fact, in addition to the time trend, which has a t statistic over three, only afdec6 has a t statistic bigger than one in absolute value. Accounting for a linear trend has important effects on the estimates.
(ii) The F statistic for joint significance of all variables except the trend and intercept, of cour) is about .54. The df in the F distribution are 6 and 123. The p-value is about .78, and so the explanatory variables other than the time trend are jointly very insignificant. We would have to conclude that once a positive linear trend is allowed for, nothing el helps to explain
log(chnimp). This is a problem for the original event study analysis.
英文歌曲经典(iii) Nothing of importance changes. In fact, the p-value for the test of joint significance of all variables except the trend and monthly dummies is about .79. The 11 monthly dummies themlves are not jointly significant: p-value≈ .59.
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C10.3 Adding log(prgnp) to equation (10.38) gives
prepop= -6.66 - .212 log(mincov t) + .486 log(usgnp t) + .285 log(prgnp t)
log()
t
(1.26) (.040) (.222) (.080)
-.027 t
(.005)
n = 38, R2 = .889, 2R = .876.
ipd
The coefficient on log(prgnp t) is very statistically significant (t statistic≈ 3.56). Becau the dependent and independent variable are in logs, the estimated elasticity of prepop with respect to prgnp is .285. Including log(prgnp) actually increas the size of the minimum wage effect: the estimated elasticity of prepop with respect to mincov is now -.212, as compared with -.169 in equation (10.38).
C10.4 If we run the regression of gfr t on pe t, (pe t-1–pe t), (pe t-2–pe t), ww2t, and pill t, the coefficient and standard error on pe t are, rounded to four decimal places, .1007 and .0298, respecti
vely. When rounded to three decimal places we obtain .101 and .030, as reported in the text.
C10.5 (i) The coefficient on the time trend in the regression of log(uclms) on a linear time trend and 11 monthly dummy variables is about -.0139 (≈ .0012), which implies that monthly unemployment claims fell by about 1.4% per month on average. The trend is very significant. There is also very strong asonality in unemployment claims, with 6 of the 11 monthly dummy variables having absolute t statistics above 2. The F statistic for joint significance of the 11 monthly dummies yields p-value≈ .0009.
(ii) When ez is added to the regression, its coefficient is about -.508 (≈ .146). Becau this estimate is so large in magnitude, we u equation (7.10): unemployment claims are estimated to fall 100[1 – exp(-.508)] ≈ 39.8% after enterpri zone designation.
(iii) We must assume that around the time of EZ designation there were not other external factors that caud a shift down in the trend of log(uclms). We have controlled for a time trend and asonality, but this may not be enough.
C10.6 (i) The regression of gfr t on a quadratic in time gives
ˆ
gfr= 107.06 + .072 t- .0080 t2
t
monday morning
(6.05) (.382) (.0051)
n = 72, R2 = .314.
翻译器中文翻英文Although t and t2 are individually insignificant, they are jointly very significant (p-value≈ .0000).
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