INSTRUCTIONS: Answer all questions in the space provided (or indicate clearly where you have continued your answer on the back of the page). Calculators are NOT permitted. Reduce all computations to the simplest form so that anyone with a calculator could attain the answer easily. Show your work and reasoning to the fullest extent possible so that part marks can be assigned as warranted. You have three hours to complete this exam. There are 25 questions (or sections) worth 5 points each. Total points = 125. Budget your time carefully. Exhibit pages should not be turned in with your exam. Remember: answer questions in a manner that reflects the econometric reasoning you have learned in this course.
1. The following questions refer to the computer output in EXHIBIT A. These are hypothetical demand data concerning a cross-sectional sample of 100 individuals and their annual consumption of frozen waffles (y). Available explanatory variables include the income of the individual in thousands of dollars per year (x) and the cost of electricity to prepare waffles in cents per dozen waffles (z). Assume that all individuals face the same price of waffles, but marginal costs of electricity vary across individuals (according to how much electricity their household consumes in total each month).
a.) Consider Regression A1 and Regression A2. If these specifications satisfied the maintained hypotheses for ordinary least squares regression, would you conclude that frozen waffles were a normal good, an inferior good, or both, depending upon the level of income. Explain your reasoning.
Adding income-squared actually reduces the adjusted R-squared value for this model. Neither x nor x-squared is individually statistically significant if both are in the model. Furthermore, the point estimates suggest that the quadratic form, if it exists at all, is a parabola that opens upwards and has a minimum, rather than a maximum, at some positive income level. There is no evidence that the positive derivative of demand with respect to income changes to a negative derivative anywhere within the sample.
b.) Regression A2
is
followed by a diagnos command. What do the results of this command
suggest, and what are the implications for your interpretation of Regression A2?
The diagnos command suggests the presence of heteroscedasticity. For each of
the range of reported tests, the null hypothesis is that the squared error from
the initial naive OLS model is NOT related to the candidate variable. The 5%
critical value for a chi-squared(1) distribution is 3.84. All but the ARCH
test value exceed this threshold (and thus have upper tail probabilities less
than 5%). Therefore, we reject the hypotheses that the OLS error variances are
independent of these factors. We do not expect to reject ARCH because this
concerns dependence of current period error variances on prior period error
variances. We wouldn't expect to see this in data which are not time-series
data. Heteroscedasticity meand that the point estimates in Regression A2 are
unbiased, but the standard errors have been computed incorrectly. None of
the inferences we might be tempted to draw from Regression A2 are valid. c.) Consider Regression
A3, Regression A4, and Regression A5. What do these regressions reveal
about the nature of any systematic variations in the magnitude of si2? If you had to choose just
one
variable that was strongly correlated with the magnitude of si2, what would it be, and why?
X and X-squared are very highly correlated in these data. Either variable by
itself (in conjunction with Z) does a good job of explaining variations in
the size of the squared regression errors. Using just X-squared with the
insignificant Z produces and R-squared of 16%, wherease just X with the
insignificant Z produces an R-squared of 14%. If we had to choose just one
variable to proxy for sigma-i-squared, we would probably choose X-squared. d.) Just prior to Regression
A6, a number of potential weighting variables are generated. Which of these
weights is more appropriate, and hence, which of
Regression A6,
Regression A7, or
Regression A8 is preferred as a "correction" for
the problems besetting the naive model in Regression
A2? Explain.
Since sigma-i-squared appears to be directly, rather than inversely,
proportional to X-squared, we should use 1/X-squared as the weighting variable,
meaning that SHAZAM will pre-process all the variables in the model by
multiplying them by 1/X. The regression will then be done
on the transformed variables, then the estimated coefficients will be reported
just as though they came from an OLS regression (in terms of the raw data).
e.) Are the standard errors for the estimated
parameters
in your preferred model smaller than those for the naive model in Regression A2? _______ For a single sample,
is it necessary that the weighted least squares estimator always produce smaller
parameter standard errors than the ordinary least squares estimator?
Explain.
Regression A7 is probably the preferred model. The standard errors of the
three coefficients are: 0.61, 1.56, and 23.9. For Regression A2, the standard
errors of the corresponding estimates are 0.77, 1.78, and 29.86. Thus, the
WLS standard errors are, in this case, smaller than the uncorrected OLS
standard errors. The relevant theoretical result is that ON AVERAGE, over all
possible samples that can be drawn from the population, the standard deviation of
the WLS estimates (if we had the true sigma-i-squared values), will be less than the
standard deviation in the OLS estimates. Here, we are comparing, for just a
single sample, just one estimate for each of these parameter dispersion
parameters, and the OLS estimate is not even calculated correctly. Thus, we
would not be surprised if the "expected" smaller parameter standard errors
from WLS did not materialize. f.) Under what circumstances will the use of a
log-log
version of your model be a remedy for heteroscedasticity? Explain.
A log-log or log-linear transformation can completely eliminate heteroscedasticity
if that the data exhibit exactly the kind of heteroscedasticity that a log-
transformation would eliminate. We saw an example in lab where a log-log
transformation rendered the transformed data model linear and homoscedastic,
whereas the levels displayed a curvilinear relationship that had heteroscedasticity.
The data set was called gourmet.dat and the two
variables to be read in are quantities of gourmet coffee beans (beans) and
income (inc). Try plotting beans against inc, and the log(beans) against
log(inc) to see what happens. If the data are not exactly the kind that
will become linear and homoscedastic with a log-log or log-linear transformation,
then logging the variables can fail to remove all the heteroscedasticity, or
it can over-correct. It may also convert a linear relationship into a nonlinear
one. Logging is not a panacea for heteroscedasticity. g.) Consider Regression
A9. If we did not have to worry about heteroscedasticity, would you prefer
this log-log model to the linear-in-variables model in Regression A2? Why or why not?
Ignoring the heteroscedasticity problem, we can compare fully linear and log-log
models by making sure to include a / loglog option on our ols command for the
log-log model (as has been done here). This ensures that the maximized log-likelihood
includes a term to accommodate the log-transformation of Y and the maximized values
of the log-likelihood for the two models is comparable. The linear model yields a
logL of -497. The log-log model gives -494. For this sample, therefore, the
log-log model produces a better fit in terms of the joint probability of observing
the sample that actually materialized. h.) Do the different implications of the models
in Regression A2 and Regression A9 regarding the income elasticity (or
inelasticity) of demand for waffles cause you any concern? Why or why not?
Regression A2 produces an estimate of income elasticity at the means of the
data equal to about 1.1, although the point elasticity will vary continuously
along the quantity/income relationship. Regression A9 produces a constant
income elasticity estimate of 0.80. If this was a price elasticity, we would
view the distinction between elastic and inelastic demand more critically.
Since that relationship is downward sloping, the response of total expenditure
to price differs across elastic and inelastic demands. Here, since we detect
no evidence of a non-linear relationship between quantity and income, the
1.0 elasticity value is not such a big deal. Each point estimate is based on
parameters that are statistically significantly different from zero, but
without further calculations, we cannot be sure that the two elasticity
estimates are statistically significantly different from each other.
i.) Do the various point estimates of the
coefficient on z (electricity cost for heating waffles) conform with economic theory?
In all specifications, quantity demanded of frozen waffles varies inversely
with the price of electricity to cook the waffles. This electricity is,
presumably, a complementary good. Economic theory tells us that an increase
in the price of a complementary good should decrease demand for that good,
and simultaneously decrease demand for waffles, which are consumed inconjunction
with the good whose price is higher. Thus the cross-price elasticity of demand
for frozen waffles with respect to the price of electricity to heat them is
negative. If we had been talking about the price of a substitute in consumption,
the cross-price effect would be expected to be positive.
2. The following questions pertain to EXHIBIT B. These are real data, and we will explore a preliminary model to explain the observed monthly time-series variation in private school construction expenditures. The variables read by the program are defined as follows:
a.) According to the results of Descriptive Statistics B and Regression B1, does new construction of private schools mimic new construction of public schools, or are private schools built when public school construction is inadequate? Explain.
There is a roughly 93% simple correlation between public and private new construction for schools. Regression B1 attempts to explain private using public and a number of other variables. The slope estimate is on the order of 0.08, indicating that on average, public school spending that is greater by $100 means an $8 increase in private school construction spending. We cannot say anything yet about the validity of the apparently huge t-ratio on this slope coefficient (without considering serial correlation in the errors or thinking about potential endogeneity problems). So the best answer is that private school construction mimics public school construction, rather than substituting for it. If private school construction was less when new public school construction is greater, we would expect a negative slope coefficient. This doesn't happen here.
b.) According to Regression B1, is new construction of private schools growing over time, even controlling for activity levels of public school construction and for the relevant demographics of school-age children? Explain.
Considering only Regression B1, it appears that new private school spending has been growing at a rate of about 0.77 million dollars per month, after controlling for the level of public school construction and seasonal variations. The t-ratio looks huge, but we don't know if we can believe it yet because these are time-series data and positive serial correlation should be suspected, meaning that the t-ratios could be (vastly) overstated.
c.) According to Regression B1, is there seasonality in private school construction? If so, summarize in words the form of this seasonality? Is it plausible?
The left-out month is January, so the basic intercept term is the intercept for January. The coefficients on the other monthly dummies tell how much expected new private school construction differs in that month compared to it's expected level in January. For example, expected new private school construction in August is higher than in January by about $37.7 million. In February, it is lower by $3.98 million compared to January. These are the point estimates. We must reserve judgment on the statistical significance of these differences until we evaluate, and possibly correct, this model for serially correlated errors. What is the overall pattern? Private school construction is highest in the summer months, when there is good weather and most kids are out of school. It is lowest in the winter months when weather is bad and attempts to build new structures at existing campuses would disrupt classes.
d.) What is the purpose of Regression B2 and what is implied from the results of this regression for the validity of Regression B1? Why?
Regression B2 considers observations for which there is complete data for up to 12 lags of the error term from the original naive OLS regression model in Regression B1. For observatons 13 through 384, we regress current on lagged error terms. We use twelve lags because these are monthly data and the possibility of an annual (i.e. 12-period) cycle might be anticipated. Since each et and et-k is our best information about the true ut and ut-k, the slope coefficients in this model are crude point estimates of the error correlations, rho, between errors separated by k periods. Since this model suggests that several of these rho coefficients may be non-zero, it seems we must tackle the problem of correcting for serially correlated errors.
e.) What is the main qualitative difference between the results from Regression B1 and Regression B3?
The biggest difference seems to be that the coefficient on P3, which used to appear strongly statistically significantly different from zero, is no longer statistically significant at the 5% level. In Regression B1, it appeared that an increase of 1000 in the 10-14 year-old-population decreased private school construction by a statistically significant $12,400. Now, it seems we cannot reject the hypothesis that the size of this cohort has no statistically significant effect on new private school construction. Other estimated coefficients are generally less significant than before, but still sufficiently significant to allow us to reject the zero hypotheses.
f.) Consider Regression B4. Is the extent to which private school construction mimics public school construction increasing or decreasing over time? Explain.
The "extent to which private school construction mimics public school construction" could be characterized algebraically as the derivative of private with respect to public. In Regression B4, this derivative is given by the coefficient on public plus t times the coefficient on tpublic. The derivative therefore changes over time according to the coefficient on tpublic = public*t. In words, a one-unit change in t (namely a one-month change in t) changes the derivative of private with respect to public by a statistically significant 0.000798. Over time, private school construction is becoming more "responsive" to public school construction levels.
g.) In Regression B4, the estimated coefficient on the T variable (the time trend variable) is no longer statistically significantly different from zero. Thus we cannot say that private school expenditure is changing systematically over time. It is statistically constant. True, False, Uncertain? Explain.
This is an invitation to confuse a slope coefficient with a derivative. In a straightforward linear model, where there are no quadratic terms or interaction terms, a slope is a derivative. Not here, however. Ther derivative of private with respect to time is not just the coefficient on T, but the coefficient on T plus PUBLIC times the coefficient on tpublic=t*public. To test whether this derivative is zero, we need a joint test of the statistical significance of the coefficients on BOTH t and tpublic. No such explicit test is provided following the regression, but we know the answer. Since the coefficient on tpublic is individually statistically significant from zero, it is certain that we can reject the hypothesis that BOTH this coefficient and that on t are simultaneously equal to zero. The statement is false.
h.) In Regression B3 and Regression B4, why is an ORDER=12 option chosen for the AUTO command?
This should be a cheap answer. Since we are dealing with monthly data, and many things have annual cycles, we should be aware of the possiblity of these cycles creating 12th order autocorrelated errors. We will usually look at least this far, rather than starting with an AR(1) model and neglecting this other likely periodicity in the data. First order serial correlation might turn out to be the most significant pattern, but we expect higher order relationships ex ante. With quarterly data, we would automatically look for AR(4) patterns; with daily data, maybe AR(7), and so on.
i.) Describe briefly what goes on "behind the scenes" when you ask SHAZAM to estimate your regression parameters using the AUTO command, rather than the OLS command.
Consider an AR(1) model, where the presumption is that ut = rho ut- 1 + et. SHAZAM comes up with an initial estimate of the correlation between current and lagged error terms and uses this to transform the regression equation by taking each variable (including the constant term), lagging it, and calculating the transformed variables according to the current-period value minus rho times the lagged-1-period value. SHAZAM then regresses uses these generalized differences in a regression where the error term is "fine" because the current minus rho-times-lagged error term is just et, which is pure noise and therefore fits the criteria for OLS. Next, SHAZAM takes the point estimates from this generalized difference regression and applies them to the raw X data to compute a fitted Y value for each observation. The difference between this fitted Y and the actual Y is a new estimate for et. If you take the correlation between this new et and its lagged value, you get an revised estimate of rho. Use this again in creating the generalized differences. Continue these iterations until some convergence criterion is achieved (either a stable largest-achievable maximized log-likelihood, or a stable smallest-achievable sum of squared errors). These final "fine-tuned" parameter estimates (intercept, slope, rho value(s)...and sigma-squared, of course) can then be reported along with their asymptotic standard errors and associated asymptotic t-ratios that allow us to test statistically the null hypotheses that individual rho parameters are zero.
3. In addition to causing flood damage, El Nino has created social costs by interfering with Southern Californians' enjoyment of their public beaches. Suppose the Department of Beaches has asked you to assess the welfare effects of beach closures due to storm-drain runoff from El Nino events. To do this, you need to estimate a model of local demand for public beaches, and then see how consumer's surplus from beach trips changes as this demand function shifts according to the number of days of beach closures (CLOSURES) each month. You collected survey data each month from a different random sample of Angelenos concerning the number of beach trips (TRIPS) they have made in that month as a function of the distance (DIST) they live from the beach. (Since beach access is free in most of Southern California, you will use this distance times average-travel-cost-per-mile as a rough proxy for the price of access). In the process of analyzing the effects of closures, you model demand by regressing TRIPS on DIST and CLOSURES and a set of sociodemographic characteristics such as age, income, and gender. Suppose a 1% change in DIST corresponds roughly to a 1% change in the "price" of a beach visit. Why should you be cautious about taking the results from this regression at face value (especially those concerning the price elasticity of demand for beach visits)? By analogy to one of the examples used in lecture, discuss a potential problem with this model and assess its implications for the estimates produced.
This was meant to remind you of the classroom example of endogenous regressors. In the classroom case, I used an example of trips to a fishing lake as a function of travel costs for access, and the goal was to do a welfare analysis of the lost consumer surplus of the lake was closed due to a chemical spill or some other environmental insult. This is merely an analogous story for an example that is closer to home (and more topical). The think to focus on is the potential endogeneity of the "price" variable in this economic analysis. Proximity to the beach in a person's choice of where to live may be strongly influenced by their affinity for beach-going (surfing, volleyball, tanning, etc.). If this is the case, then endogeneity bias could afflict the estimated coefficient on the price (distance) variable above. Suppose, as expected, greater distance from the beach (higher cost of access) decreases expected number of trips. But we cannot control for interest in beach-going; greater distance from the beach also captures lesser interest in beach-going, which leads to even fewer trips to the beach. The negative coefficient on DIST will be even more negative than it ought to be, because people can choose where they live. We will perceive a greater-than-actual responsiveness of beach visits to distance, and thus we will be overestimating the magnitude of the price elasticity (since these are usually treated as absolute values).
4. If your dependent variable is a (0,1) dummy
variable that indicates the category to which an observation belongs, you probably
want to try a PROBIT or a LOGIT method for estimating your model. How does the
interpretation of set of PROBIT or LOGIT results differ from the
interpretation of the analogous set of OLS results?
The estimates produces by a probit or logit model that look like slopes and intercepts in the output are actually standardized slopes and intercepts, known only up to a scale factor (i.e. in ratio to the unknown error standard deviation). So the size of these coefficients is not viewed in the same way (as the change in Y for a one-unit change in X). Instead, we focus on the sign and the statistical significance of these coefficients, in particular, the "slopes." If a "slope" is positive, it means that an increase in the corresponding X variable increases the latent propensity to choose the 1 alternative, thereby also increasing the predicted probability of choosing the 1 alternative. (Analogously for negative slope coefficients.) If the fitted probability is greater than 0.5, we predict the individual will choose 1. If it is less than 0.5, we predict they will choose 0. The asymptotic t-ratios allow us to test zero hypotheses about these "slopes." If the t-ratios exceed (roughly) 2 in absolute value, we can reject the hypothesis that the relevant X variable has no effect on fitted choice probabilities.
5. Suppose you have been hired by a large national
recreational equipment cooperative to assess individual consumer expenditures on
the types of products sold by the cooperative. You are provided with some survey
data on individual expenditures (EXP) by AGE, gender (FEM=1 if female) and income
(INC, in thousands of dollars per year). The best-fitting model you discover is
displayed in Exhibit C.
a.) Based on the point estimates, provide a formula that would give expected expenditures for a randomly selected female. (Two significant digits will be adequate.)
E[Y] = (129.8-39.1) + (4.96 - 6.08) AGE + (-0.063 + 0.079) AGE2
- 0.575 INC + 0.027 AGE*INC
b.) Explain how you would go about testing whether expected expenditures differ by gender.
The only way we could have no difference in expected expenditures by gender would be if the coefficients on FEM, FEMAGE, and FEMAGE2 were all simultaneously zero. Thus, we would need to test the joint statistical significance of these coefficients with an appropriate F-test. None is provided, but we can be sure it would reject the jointly zero hypothesis because two of these coefficients are individually statistically significant.
c.) What appears to be the main difference between the male and female age profiles of expenditure on recreational equipment in this sample?
We have fit quadratic age profiles. For the males, we zero-out the terms involving FEM (FEM, FEMAGE, and FEMAGE2). The male age profile has a positive slope at AGE=0 and can be depicted as a parabola opening downward. We could solve for the age at which male expenditures on recreational equipment peak within the range of the data. In contrast, the female age profile has a negative slope at AGE=0 and a positive coefficient on AGE2, so that it can be sketched as a parabola opening upward, having a minimum within the range of the sample. We could solve for the location of this minimum, but you've done that for credit before.
d.) Does an extra $1000 of annual income have any statistically discernible effect on recreational equipment expenditures? Explain carefully.
Income is measured in thousands of dollars, so we are talking about a "1-unit" change. The coefficient on income alone is not statistically significant, but the question concerns a derivative of recreational equipment expenditures with respect to income. This derivative is -0.575 + 0.027*age. This derivative cannot be zero if 0.027 is statistically significantly different from zero (unless we are talking only about people with AGE=0...an unlikely prospect).
6. In a regression model like that used for the time- series data in Exhibit B, can there be any multicollinearity among the dummy variables for each month? Explain your answer. If none of the individual t-test statistics on these monthly dummy variables is individually significantly different from zero, is it likely to be necessary to perform an F-test to assess the possible joint significance of the full set of dummies? Explain.
BONUS (4 points): One point for each section with anything
plausible.
b.) For which topics would you have preferred less class coverage, and for which would you have preferred more?
c.) Which types of lab sessions did you find most useful to your understanding of the course material (if you remember any of them at all)? (Alternately, of the 8 lab sessions, how many were you able to attend?)
d.) For which topics might new WWWeb-based visualization aids be most helpful? (Consider the set of JAVA applets accessible from the current web site.)
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