Wooldridge 6e Ch10 IM

137
CHAPTER 10
Basic Regression Analysis with Time Series Data
Table of Contents
Teaching notes 138

Solutions to Problems 139
Solutions to Computer Exercises 142
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138
TEACHING NOTES
Because of its realism and its care in stating assumptions, this chapter puts a somewhat heavier
burden on the instructor and the student than traditional treatments of time series 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 series data that are not present for cross-sectional
applications. Trends, seasonality, and high persistence are ubiquitous in time series data. By
this time, students should have a firm grasp of multiple regression mechanics and inference. So,
you can focus on those features that make time series applications different from cross-sectional
ones.
I think it is useful to discuss static and finite distributed lag models at the same time, as these at
least have a shot at satisfying the Gauss-Markov assumptions. Many interesting examples have
distributed lag dynamics. In discussing the time series 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
unbiased – as opposed 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 serial correlation
assumption. As the absence of serial 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 unbiasedness 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 because they arise in many time series examples.
A novel feature of the text is the discussion of how to compute goodness-of-fit measures with a
trending or seasonal dependent variable. While detrending or deseasonalizing y is hardly perfect
(and does not work with integrated processes), it is better than simply reporting the very high R-
squareds that often come with time series regressions with trending variables.
139
SOLUTIONS TO PROBLEMS
10.1 (i) Disagree. Most time series processes are correlated over time, and many of them are
strongly correlated. This means they cannot be independent across observations, which simply
represent different time periods. Even series that do appear to be roughly uncorrelated – such as
stock returns – do not appear to be independently distributed, as you will see 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 serial correlation assumptions.
(iii) Disagree. Trending variables are used all the time as dependent variables in a regression
model. We do need to be careful in interpreting the results because we may simply find a
spurious association between y t and trending explanatory variables. Including a trend in the
regression is a good idea for trending dependent or independent variables. As discussed 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 represents a year and is not associated with
any season.
10.2 We follow the hint and write
gGDP t-1 = α 0 + δ 0 int t-1 + δ 1 int t-2 + u t-1 ,
and plug this into the right-hand-side of the int t equation:
int t = γ 0 + γ 1 (α 0 + δ 0 int t-1 + δ 1 int t-2 + u t-1 – 3) + v t

= (γ 0 + γ 1 α0 – 3γ1) + γ1δ0intt-1 + γ1δ1intt-2 + γ1ut-1 + vt.
Now, by assumption, ut-1 has zero mean and is uncorrelated with all the right-hand-side variables
in the previous equation, except itself, of course. So,
Cov(int,ut-1) = E(intt ⋅ ut-1) = γ1E( ut −1 ) > 0

2
because γ1 > 0. If σ u = E( ut ) for all t, then Cov(int,ut-1) = γ1 σ u . This violates the strict
2 2 2
exogeneity assumption, TS.2. While ut is uncorrelated with intt, intt-1, and so on, ut is correlated
with intt+1.
10.3 Write
y* = α0 + (δ0 + δ1 + δ2)z* = α0 + LRP ⋅ z*,
and take the change: ∆y* = LRP ⋅ ∆z*.
140
10.4 We use the R-squared form of the F statistic (and ignore the information on R 2 ). 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
log(hsestrtst) = α0 + α1t + δ1Q2t + δ2Q3t + δ3Q4t + β1intt +β2log(pcinct) + ut,
where Q2t, Q3t, and Q4t are quarterly dummy variables (the omitted quarter is the first) and the
other variables are self-explanatory. This inclusion of the linear time trend allows the dependent
variable and log(pcinct) to trend over time (intt probably does not contain a trend), and the
quarterly dummies allow all variables to display seasonality. The parameter β2 is an elasticity
and 100 ⋅ β1 is a semi-elasticity.
10.6 (i) Given δj = γ0 + γ1 j + γ2 j2 for j = 0,1,  ,4, we can write
yt = α0 + γ0zt + (γ0 + γ1 + γ2)zt-1 + (γ0 + 2γ1 + 4γ2)zt-2 + (γ0 + 3γ1 + 9γ2)zt-3

+ (γ0 + 4γ1 + 16γ2)zt-4 + ut
= α0 + γ0(zt + zt-1 + zt-2 + zt-3 + zt-4) + γ1(zt-1 + 2zt-2 + 3zt-3 + 4zt-4)
+ γ2(zt-1 + 4zt-2 + 9zt-3 + 16zt-4) + ut.
(ii) This is suggested in part (i). For clarity, define three new variables: zt0 = (zt + zt-1 + zt-2 +
zt-3 + zt-4), zt1 = (zt-1 + 2zt-2 + 3zt-3 + 4zt-4), and zt2 = (zt-1 + 4zt-2 + 9zt-3 + 16zt-4). Then, α0, γ0, γ1,
and γ2 are obtained from the OLS regression of yt on zt0, zt1, and zt2, where t = 1, 2,  , n.
(Following our convention, we let t = 1 denote the first time period where we have a full set of
regressors.) The δˆ can be obtained from δˆ = γˆ + γˆ j + γˆ j2.
j j 0 1 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 on 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, Rur , from the regression of yt on zt, zt-1,  , zt-4 and the restricted R-
2
2
squared from the regression in part (ii), Rr . The F statistic is
141
( Rur2 − Rr2 ) (n − 6)
=F ⋅ .
(1 − Rur2 ) 2
Under H0 and the CLM assumptions, F ~ F2,n-6.
10.7 (i) pet-1 and pet-2 must be increasing by the same amount as pet.
(ii) The long-run effect, by definition, should be the change in gfr when pe increases
permanently. But a permanent increase means the level of pe increases and stays at the new
level, and this is achieved by increasing pet-2, pet-1, and pet 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, ut, is uncorrelated with the regressors in
every time period: current, past, and future. Sequential exogeneity states that ut is uncorrelated
with current and past regressors, so it is implied by strict exogeneity. In terms of conditional
means, strict exogeneity is E(ut | ..., xt −1 , xt , xt +1 ,...) = 0 . So, ut conditional on any subset of
(..., xt −1 , xt , xt +1 ,...) , including (xt , xt −1 ,...) , also has a zero conditional mean. But the latter
condition is the definition of sequential exogeneity.
(ii) Sequential exogeneity implies that ut is uncorrelated with xt, xt-1, … , which, of course,
implies that ut is uncorrelated with xt (which is contemporaneous exogeneity stated in terms of
zero correlation). In terms of conditional means, E(ut | xt , xt −1 ,...) = 0 implies that ut has zero
mean conditional on any subset of variables in (xt , xt −1 ,...) . In particular, E(ut | xt ) = 0 .
(iii) No, OLS is not generally unbiased under sequential exogeneity. To show unbiasedness,
we need to condition on the entire matrix of explanatory variables, X, and use E(ut | X) = 0 for
all t. But this condition is strict exogeneity, and it is not implied by sequential exogeneity.
(iv) The model and assumption imply
E(ut | pccont , pccont −1 ,...) = 0 ,
which means that pccont is sequentially exogenous. (One can debate whether three lags are
enough to capture the distributed lag dynamics, but the problem asks you to assume this.) But
pccont may very well fail to be strictly exogenous because of feedback effects. For example, a
shock to the HIV rate this year – manifested as ut > 0 – could lead to increased condom usage in
the future. Such a scenario would result in positive correlation between ut and pccont+h for h > 0.
OLS would still be consistent, but not unbiased.
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SOLUTIONS TO COMPUTER EXERCISES
C10.1 Let post79 be a dummy variable equal to one for years after 1979, and zero otherwise.
Adding post79 to equation 10.15) gives
i3t = 1.30 + .608 inft + .363 deft + 1.56 post79t

(0.43) (.076) (.120) (0.51)
n = 56, R2 = .664, R 2 = .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( chnimp) = −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, R 2 = .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
course) is about .54. The df values in the F distribution are 6 and 123. The p-value is about .78,
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 else helps to explain
log(chnimp). This is a problem for the original event study analysis.
(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
themselves are not jointly significant: p-value ≈ .59.
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C10.3 Adding log(prgnp) to equation (10.38) gives

log( prepopt ) = −6.66 − .212 log(mincovt) + .486 log(usgnpt) + .285 log(prgnpt)
(1.26) (.040) (.222) (.080)
− .027 t
(.005)
n = 38, R2 = .889, R 2 = .876.
The coefficient on log(prgnpt) is very statistically significant (t statistic ≈ 3.56). Because the
dependent and independent variables are in logs, the estimated elasticity of prepop with respect
to prgnp is .285. Including log(prgnp) actually increases 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 gfrt on pet, (pet-1 – pet), (pet-2 – pet), ww2t, and pillt, the
coefficient and standard errors on pet, rounded to four decimal places, are .1007 and .0298,
respectively. 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 (se ≈ .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 seasonality 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 (se ≈ .146). Because this
estimate is so large in magnitude, we use equation (7.10): unemployment claims are estimated to
fall 100[1 – exp(−.508)] ≈ 39.8% after enterprise zone designation.
(iii) We must assume that around the time of EZ designation there were not other external
factors that caused a shift down in the trend of log(uclms). We have controlled for a time trend
and seasonality, but this may not be enough.
C10.6 (i) The regression of gfrt on a quadratic in time gives
ˆ = 107.06 + .072 t − .0080 t2

gfrt
(6.05) (.382) (.0051)
n = 72, R2 = .314.
Although t and t2 are individually insignificant, they are jointly very significant (p-value ≈
.0000).
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 as the dependent variable in (10.35) gives R2 ≈ .602, compared with about .727
(ii) Using gfr t
if we do not initially detrend. Thus, the equation still explains a fair amount of variation in gfr
even after we net out the trend in computing the total variation in gfr.
(iii) The coefficient and standard errors on t3 are about −.00129 and .00019, respectively,
which results in a very significant t statistic. It is difficult to know what to make of this. The
cubic trend, like the quadratic, is not monotonic. So, this almost becomes a curve-fitting
exercise.
C10.7 (i) The estimated equation is
 = .0081 + .571 gyt

gc t
(.0019) (.067)
n = 36, R2 = .679.
This equation implies that if income growth increases by one percentage point, consumption
growth increases by .571 percentage points. The coefficient on gyt is very statistically significant
(t statistic ≈ 8.5).
(ii) Adding gyt-1 to the equation gives
 = .0064 + .552 gyt + .096 gyt-1

gc t
(.0023) (.070) (.069)
n = 35, R2 = .695.
The t statistic on gyt-1 is only about 1.39, so it is not significant at the usual significance levels.
(It is significant at the 20% level against a two-sided alternative.) In addition, the coefficient is
not especially large. At best, there is weak evidence of adjustment lags in consumption.
(iii) If we add r3t to the model estimated in part (i), we obtain
 = .0082 + .578 gyt − .00021 r3t

gc t
(.0020) (.072) (.00063)
n = 36, R2 = .680.
The t statistic on r3t is very small. The estimated coefficient is also practically small: a one-
point increase in r3t reduces consumption growth by about .021 percentage points.
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 = 92.05 + .089 pet − .0040 pet-1 + .0074 pet-2 + .018 pet-3

gfr + .014 pet-4
t
(3.33) (.126) (.1531) (.1651) (.154) (.105)
− 21.34 ww2t − 31.08 pillt
(11.54) (3.90)
n = 68, R2 = .537, R 2 = .483.
The p-value for the F statistic of joint significance of pet-3 and pet-4 is about .094, which is very
weak evidence against H0.
(ii) The LRP and its standard error can be obtained as the coefficient and standard error on
pet in the regression
gfrt on pet, (pet-1 – pet), (pet-2 – pet), (pet-3 – pet), (pet-4 – pet), ww2t, pillt.
 ≈ .124 (se ≈ .030), which is above the estimated LRP with only two lags (.101).
We get LRP
The standard errors are the same rounded to three decimal places.
(iii) We estimate the PDL with the additional variables ww2t and pillt. To estimate γ0, γ1, and
γ2, we define the variables
z0t = pet + pet-1 + pet-2 + pet-3 + pet-4

z1t = pet-1 + 2pet-2 + 3pet-3 + 4pet-4
z2t = pet-1 + 4pet-2 + 9pet-3 + 16pet-4.
Then, run the regression gfrtt on z0t, z1t, z2t, ww2t, pillt. Using the data in FERTIL3.RAW gives
(to three decimal places) γˆ0 = .069, γˆ1 = –.057, γˆ2 = .012. So, δˆ0 = γˆ0 = .069, δˆ1 = .069 −
.057 + .012 = .024, δˆ2 = .069 – 2(.057) + 4(.012) = .003, δˆ3 = .069 – 3(.057) + 9(.012) = .006, δˆ4
= .069 – 4(.057) + 16(.012) = .033. Therefore, the LRP is .135. This is slightly above the .124
obtained from the unrestricted model, but not much.
Incidentally, the F statistic for testing the restrictions imposed by the PDL is about [(.537 −
.536)/(1 − .537)](60/2) ≈ .065, which is very insignificant. Therefore, the restrictions are not
rejected by the data. Anyway, the only parameter we can estimate with any precision, the LRP,
is not very different in the two models.
C10.9 (i) The sign of β 2 is fairly clear-cut: as interest rates rise, stock returns fall, so β 2 < 0.
Higher interest rates imply that T-bill and bond investments are more attractive and also signal a
future slowdown in economic activity. The sign of β1 is less clear. While economic growth can
146
be a good thing for the stock market, it can also signal inflation, which tends to depress stock
prices.
(ii) The estimated equation is

rsp 500 t = 18.84 + .036 pcipt −1.36 i3t
(3.27) (.129) (0.54)
n = 557, R2 = .012.
A one percentage point increase in industrial production growth is predicted to increase the stock
market return by .036 percentage points (a very small effect). On the other hand, a one
percentage point increase in interest rates decreases the stock market return by an estimated 1.36
percentage points.
(iii) Only i3 is statistically significant with t statistic ≈ −2.52.
(iv) The regression in part (i) has nothing directly to say about predicting stock returns
because the explanatory variables are dated contemporaneously with rsp500. In other words, we
do not know i3t before we know rsp500t. What the regression in part (i) says is that a change in
i3 is associated with a contemporaneous change in rsp500.
C10.10 (i) The sample correlation between inf and def is only about .098, which is pretty small.
Perhaps, surprisingly, inflation and the deficit rate are practically uncorrelated over this period.
Of course, this is a good thing for estimating the effects of each variable on i3, as it implies
almost no multicollinearity.
(ii) The equation with the lags is
i3t = 1.61 + .343 inft + .382 inft-1 − .190 deft + .569 deft-1
(0.40) (.125) (.134) (.221) (.197)
n = 55, R2 = .685, R 2 = .660.
(iii) The estimated LRP of i3 with respect to inf is .343 + .382 = .725, which is somewhat
larger than .606, which we obtain from the static model in (10.15). But the estimates are fairly
close considering the size and significance of the coefficient on inft-1.
(iv) The F statistic for significance of inft-1 and deft-1 is about 5.22, with p-value ≈ .009. So,
they are jointly significant at the 1% level. It seems that both lags belong in the model.
C10.11 (i) The variable beltlaw becomes one at t = 61, which corresponds to January 1986. The
variable spdlaw goes from zero to one at t = 77, which corresponds to May 1987.
(ii) The OLS regression gives
147

log(totacc) = 10.469 + .00275 t − .0427 feb + .0798 mar + .0185 apr
(.019) (.00016) (.0244) (.0244) (.0245)
+ .0321 may + .0202 jun + .0376 jul + .0540 aug

(.0245) (.0245) (.0245) (.0245)
+ .0424 sep + .0821 oct + .0713 nov + .0962 dec

(.0245) (.0245) (.0245) (.0245)
n = 108, R2 = .797.
When multiplied by 100, the coefficient on t gives roughly the average monthly percentage
growth in totacc, ignoring seasonal factors. In other words, once seasonality was eliminated,
totacc grew by about .275% per month over this period, or 12(.275) = 3.3% at an annual rate.
There is pretty clear evidence of seasonality. Only February has a lower number of total
accidents than the base month, January. The peak is in December: roughly, there are 9.6% more
accidents in December over January in the average year. The F statistic for joint significance of
the monthly dummies is F = 5.15. With 11 and 95 df, this gives a p-value essentially equal to
zero.
(iii) I will report only the coefficients on the new variables:

log(totacc) = 10.640 + … + .00333 wkends − .0212 unem
(.063) (.00378) (.0034)
− .0538 spdlaw + .0954 beltlaw

(.0126) (.0142)
n = 108, R2 = .910.
The negative coefficient on unem makes sense if we view unem as a measure of economic
activity. As economic activity increases – unem decreases – we expect more driving and
therefore more accidents. The estimate that a one percentage point increase in the
unemployment rate reduces the total accidents by about 2.1%. A better economy does have costs
in terms of traffic accidents.
(iv) At least initially, the coefficients on spdlaw and beltlaw are not what we might
expect. The coefficient on spdlaw implies that accidents dropped by about 5.4% after the
highway speed limit was increased from 55 to 65 miles per hour. There are at least a couple of
possible explanations. One is that people became safer drivers after the speed limit was
increased, recognizing that they must be more cautious. It could also be that some other change
– other than the increased speed limit or the relatively new seat belt law – caused lower total
number of accidents, and we have not properly accounted for this change.
148
The coefficient on beltlaw also seems counterintuitive at first. But, perhaps, people became less
cautious once they were forced to wear seatbelts.
(v) The average of prcfat is about .886, which means, on average, slightly less than one
percent of all accidents result in a fatality. The highest value of prcfat is 1.217, which means
there was one month where 1.2% of all accidents resulted in a fatality.
(vi) As in part (iii), I do not report the coefficients on the time trend and seasonal dummy
variables:

prcfat = 1.030 + … + .00063 wkends − .0154 unem
(.103) (.00616) (.0055)
+ .0671 spdlaw − .0295 beltlaw

(.0206) (.0232)
n = 108, R2 = .717.
Higher speed limits are estimated to increase the percentage of fatal accidents by .067 percentage
points. This is a statistically significant effect. The new seat belt law is estimated to decrease
the percentage of fatal accidents by about .03, but the two-sided p-value is about .21.
Interestingly, increased economic activity also increases the percentage of fatal accidents. This
may be because more commercial trucks are on the roads, and these probably increase the chance
that an accident results in a fatality.
C10.12 (i) OLS estimation using all of the data gives
 = 1.05 + .502 unem

inf
(1.55) (.266)
n = 56, R2 = .062, R 2 = .045,
so there are 56 years of data.
(ii) The estimates are similar to those in equation (10.14). Adding the extra years does not
help in finding a tradeoff between inflation and unemployment. In fact, the slope estimate
becomes even larger (and is still positive) in the full sample.
(iii) Using only data from 1997 to 2003 gives
 = 4.16 − .378 unem

inf
(1.65) (.334)
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n = 7, R2 = .204, R 2 = .044.
The equation now shows a tradeoff between inflation and unemployment: a one percentage point
increase in unem is estimated to reduce inf by about .38 percentage points. Not surprisingly,
with such a small sample size, the estimate is not statistically different from zero: the two-sided
p-value is .31. So, while it is tempting to think that the inflation-unemployment tradeoff
reemerges in the last part of the sample, the estimates are not precise enough to draw that
conclusion.
(iv) The regressions in parts (i) and (iii) are an example of this setup, with n1 = 49 and n2 = 7.
The weighted average of the slopes from the two different periods is (49/56)⋅(.468) +
(7/56)⋅(−.378) ≈ .362. But the slope estimate on the entire sample is .502. Generally, there is no
simple relationship between the slope estimate on the entire sample and the slope estimates on
two sub-samples.

gwage232 = .0022 + .151 gmwage + .244 gcpi
(.0004) (.001) (.082)
n = 611, R2 = .293.
The coefficient on gmwage implies that a one percentage point growth in the minimum wage is
estimated to increase the growth in wage232 by about .151 percentage points.
The coefficient on gmwage is statistically significant (t statistic ≈ 15.59 and p-value is about
zero.)
(ii) When 12 lags of gmwage are added, the sum of all coefficients is about .198, which is
somewhat higher than the .151 obtained from the static regression. Plus, the F statistic for lags 1
through 12 given p-value = .058 shows they are jointly, marginally statistically significant. (Lags
8 through 12 have fairly large coefficients, and some individual t statistics are significant at the
5% level.)
(iii) The estimated equation is

gemp 232 = −.0004 − .0019 gmwage − .0055 gcpi
(.0010) (.0228) (.1938)
n = 611, R2 = .000.
The coefficient on gmwage is puny with a very small t statistic. In fact, the R-squared is
practically zero, which means neither gmwage nor gcpi has any effect on employment growth in
sector 232.
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(iv) Adding lags of gmwage does not change the basic story. The F test of joint significance
of gmwage and lags 1 through 12 of gmwage gives p-value = .439. The coefficients change sign,
and none is individually statistically significant at the 5% level. Therefore, there is little evidence
that minimum wage growth affects employment growth in sector 232, either in the short run or
the long run.
C10.14 (i) The variable approve ranges between 30.94 and 88.65. The average value of approve
is about 53.58.
(ii) The estimated equation is
� 𝑡𝑡 = 802.35 − 115.191lcpifood
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 33.117lrgasprice + .727 unemploy
(141.17) (32.494) (7.403) (1.503)
n = 78, R2 = .816.
(iii) The estimated elasticity values of approve with respect to lcpifood and lrgasprice are −
115.191 and − 33.11, respectively. The coefficient on unemploy implies that if unemployment
rate increases by one percentage point, the approval rate increases by .727 percentage points. The
signs of coefficients lcpifood and lrgasprice imply that as CPI on food and real gas price
increases, the approval rate falls. The sign of unemployment rate is unexpected since the
approval rate increases as the unemployment rate increases.
The t statistic values for lcpifood and lrgasprice are about − 3.54 and − 4.47, respectively, with a
p-value of about zero, so these two variables are significant at the usual significance levels. But
the variable unemploy, with t statistic 0.483, is highly insignificant.
(iv) The estimated equation is
� 𝑡𝑡 = 688.57 − 98.507lcpifood
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 29.361lrgasprice + 2.354 unemploy
(113.94) (26.187) (6.006) (1.261)
+ 21.527 11-Sep + 4.408 iraqinvade

(3.211) (3.155)
n = 78, R2 = .888.
The variable 11-Sep is statistically significant with t statistic 6.70 and a p-value of about zero.
But the variable iraqinvade is insignificant with t statistic 1.40 and a p-value of .167.
The coefficient on 11-Sep implies that the approval rate was higher for 09/2001 and the
following two months, and the coefficient on iraqinvade also implies that the approval rate was
increasing three months after the Iraq invasion.
151
(v) Adding the dummy variables changes the coefficient on unemploy to 2.354 and there is not
much change in the other variables. The coefficient on unemploy is hard to rationalize.
(vi) The estimated elasticity of approval rate with respect to lsp500 is 1.439. Also, it is
statistically insignificant with t statistic .178 and a p-value of .859. Therefore, the stock market
does not have an important effect on the presidential approval rating.

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137

Teaching notes 138

10.2 We follow the hint and write

gGDP t-1 = α 0 + δ 0 int t-1 + δ 1 int t-2 + u t-1 ,

and plug this into the right-hand-side of the int t equation:

int t = γ 0 + γ 1 (α 0 + δ 0 int t-1 + δ 1 int t-2 + u t-1 – 3) + v t

Cov(int,ut-1) = E(intt ⋅ ut-1) = γ1E( ut −1 ) > 0

y* = α0 + (δ0 + δ1 + δ2)z* = α0 + LRP ⋅ z*,

and take the change: ∆y* = LRP ⋅ ∆z*.

F = [(.305 − .281)/(1 − .305)](124/3) ≈ 1.43,

log(hsestrtst) = α0 + α1t + δ1Q2t + δ2Q3t + δ3Q4t + β1intt +β2log(pcinct) + ut,

10.6 (i) Given δj = γ0 + γ1 j + γ2 j2 for j = 0,1,  ,4, we can write

yt = α0 + γ0zt + (γ0 + γ1 + γ2)zt-1 + (γ0 + 2γ1 + 4γ2)zt-2 + (γ0 + 3γ1 + 9γ2)zt-3

Under H0 and the CLM assumptions, F ~ F2,n-6.

(iv) The model and assumption imply

E(ut | pccont , pccont −1 ,...) = 0 ,

SOLUTIONS TO COMPUTER EXERCISES

i3t = 1.30 + .608 inft + .363 deft + 1.56 post79t

n = 56, R2 = .664, R 2 = .644.

C10.2 (i) Adding a linear time trend to (10.22) gives

n = 131, R2 = .362, R 2 = .325.

C10.3 Adding log(prgnp) to equation (10.38) gives

n = 38, R2 = .889, R 2 = .876.

C10.6 (i) The regression of gfrt on a quadratic in time gives

ˆ = 107.06 + .072 t − .0080 t2

C10.7 (i) The estimated equation is

 = .0081 + .571 gyt

(ii) Adding gyt-1 to the equation gives

 = .0064 + .552 gyt + .096 gyt-1

(iii) If we add r3t to the model estimated in part (i), we obtain

 = .0082 + .578 gyt − .00021 r3t

C10.8 (i) The estimated equation is

 = 92.05 + .089 pet − .0040 pet-1 + .0074 pet-2 + .018 pet-3

n = 68, R2 = .537, R 2 = .483.

z0t = pet + pet-1 + pet-2 + pet-3 + pet-4

(ii) The estimated equation is

(iii) Only i3 is statistically significant with t statistic ≈ −2.52.

(ii) The equation with the lags is

(ii) The OLS regression gives

+ .0321 may + .0202 jun + .0376 jul + .0540 aug

+ .0424 sep + .0821 oct + .0713 nov + .0962 dec

(iii) I will report only the coefficients on the new variables:

− .0538 spdlaw + .0954 beltlaw

+ .0671 spdlaw − .0295 beltlaw

C10.12 (i) OLS estimation using all of the data gives

 = 1.05 + .502 unem

n = 56, R2 = .062, R 2 = .045,

so there are 56 years of data.

(iii) Using only data from 1997 to 2003 gives

 = 4.16 − .378 unem

C10.13 (i) The estimated equation is

(iii) The estimated equation is

(ii) The estimated equation is

(iv) The estimated equation is

+ 21.527 11-Sep + 4.408 iraqinvade

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