UNIVERSITY OF CALIFORNIA, LOS ANGELES

Department of Economics

Economics 143 - Midterm Key (Outline)

• 1. U.S. hypothetical sample.
  • a.) Fill in the blanks:
    • - The quantity data for each city is in bus trips per capita for that city. The output of the STAT command in Exhibit 1 provides the mean value of Q. The answer is 4.4824 trips per capita.
    • - The maximum single-zone bus fare $1.50, again from the output of the STAT command.
    • - The standard deviation in per-capita disposable income is 1.8572 thousand dollars per year. Intuition suggests the units, because of the magnitude of the mean disposable income being 19.706. Clearly, this can't be plain dollars.
    • - The descriptive statistics you have just provided refer to the marginal distributions of Q, P, and Y.
    • - The correlation between quantity and price in this sample is - 0.41686. The two variables are negatively related.
    • - There are no units for this correlation measure because correlation is a unit-free entity. It is formed by dividing covariance (which is in units of trips*dollars) by the product of the marginal standard deviations of Q and P (which are measured in units of trips and units of dollars). All the units cancel.
  • b. Consider Regression A1. If the transit authority had chosen to rely only on its data on trips and fares:
    • - The point estimate of the effect of a one-unit increase in bus fares on the expected number of per-capita bus trips (on average, across these 17 cities), would have been -3.3775. This does make sense, given typical numbers of bus trips, since trips range from 1.5 to 7.1 in the sample. I asked this question because it is important to compare parameter point estimates with your common sense to determine whether the empirical model you have just estimated agrees with your intuitive "theory." In this case, the results probably do agree with our intuition. This illustrates that passing the so-called "laugh test" with an estimated model does not necessarily mean the model is right. We see later that the price coefficient is afflicted by serious omitted variables bias. But this could go undetected if we didn't challenge the data further to see what other relationships might be present. Who knows how many published research papers draw incorrect conclusions because of omitted variables? Who knows how many major policy decisions are made based on faulty evidence such as this? It is a sobering thought.
    • - The point estimate of the effect of a ten-cent increase in bus fares on trips (on average, across these 17 cities) would have been one- tenth this size, since we would be considering the effect of a 1/10 unit increase in P on Q. People would be expected to take about .34 fewer trips. This doesn't seem inconsistent with intuition either.
    • - The intercept is the expected number of bus trips if the fare is zero. This would be demand for bus trips at a zero price. Since it is unlikely that bus fares would ever be zero (and the smallest value of P we observe in our sample is $0.85), the intercept is really just an artifact of our choice of a linear functional form, projected back to where it crosses the vertical axis.
  • c.)If demand is completely inelastic, this means that quantity demanded with not change at all if price changes. This goes along with a Q that is unaffected by changes in P in our story. This translates into a zero slope on the P variable. A zero slope is tested easily by considering the t-ratio on the P variable in Regression A1. The t-ratio is only -1.776. With 15 degrees of freedom, the 5% critical value of the t-distribution is plus or minus 2.131, so the coefficient is not statistically significantly different from zero at this level. (Extra: At the 10% level of significance, the critical values are plus or minus 1.753, so this test statistic would only just barely reject the null hypothesis at the 10% level.) At the usual 5% level, however, the activist's assertion is not rejected. With zero elastic demand, the claim about the effect of the 20% increase inn fares would be true.
  • d.) Based on Regression A1, a point prediction about the likely average weekly ridership figures if they set fares at $1.20 would be calculated by plugging this price into the fitted regression equation: 8.5155-3.3775*1.20. Furthermore, they want a likely range of ridership levels that would result. This would mean a confidence interval for the true value of Q when P equals this particular value. You would use the formula (given on the crib sheet) for a confidence interval for B1 + B2*1.20. For a complete answer, you would indicate where each of the ingredient terms could be found. The value of n is 17, the value of X0 is 1.20, the value of X-bar is mean price (= 1.1941), the value of s is the standard error of the estimate, 1.3438, and the value of the sum of the "little x-deviations squared" is found by using the value of "variance of the estimate-sigma**2" combined with the square of the standard error on the slope coefficient (which has the desired term in its denominator). We worked through this in class.
  • e.)In Regression A2, you include your additional income data corresponding to each observation. The point estimate of the effect of a one-unit increase in bus fares on per-capita weekly ridership goes from -3.3775 to - 22.403 as you move to this more-general model. Since trips range only from 1.5 to 7.1 in the data, this seems a awfully extreme. But notice that prices only vary from $0.85 to $1.50 in the sample, so a one-unit (one dollar) change in the price would be huge. Plausible sized changes in prices are not going to be on the order of $1.00. They are probably going to be smaller. This says, for example, that a $0.10 increase in the fare would decrease average trips by 2.24 per week. [Bear in mind that this is an imaginary data set. This is not necessarily the way the world works, but it could be.] How would these people get around if they are not taking busses? Walking, bicycling, changing jobs, or quitting jobs since they can't afford bus fare any longer. Demand elasticity in part belies the availability of close substitutes.
  • f.) If we now test the activist's assertion based on Regression A2, we find a clear rejection of the null hypothesis of a zero slope on the price variable (at any level of significance). A t-test statistic value of -7.661 is extremely improbable if the true slope in the population is zero. Thus we reject the null hypothesis. The reason the effect of price was mostly obscured (and greatly biased) in the first regression model is that bus fares appear to be correlated strongly with incomes across cities. The STAT output confirms that this correlation is 0.94678. Higher prices for bus trips were correlated with higher incomes of bus passengers in this sample, so the expected decrease in quantity demanded could not be seen (it was offset by increases in demand due to higher incomes). Only when both variables are included separately can we disentangle their distinct effects on quantities demanded (in these hypothetical data). So a 20% increase in bus fares would not mean that inner-city residents would shell out a total of 20% more in transportation costs.
  • g.) Microeconomics tells us that the price elasticity of demand reveals whether a price increase will make revenues larger or smaller. Price elasticity is (% change in Q)/(% change in P), which is roughly equal to (delta Q/Q)/(delta P/P) = (delta Q/delta P)*(P/Q). Recall that a linear demand curve does not have constant elasticity, so we generally "measure" elasticity at the means of the data. In Exhibit A would you find (delta Q/delta P) as the slope on the price variable in the multiple regression (= -22.403), sample average P and sample average Q can be read off the results of the STAT command? SHAZAM is indeed smart enough to anticipate your interest in these elasticities "at the means of the data." They appear in the last column of the core regression output. Here, the answer is -5.9684. Demand at the means of the data would be highly elastic. Thus an increase in price would result in a decrease in total revenues for the bus company (and a decrease in aggregate costs of public transit for this group of consumers). Why? Because they substitute away from bus transit very heavily when the price changes. NOTE that this contrasts with the "finding" from Regression A1, where the elasticity appeared to be only -0.8998 which is inelastic, implying that an increase in price would increase total revenues (and hence aggregate costs for consumers). Incidentally, the apparent fact that a 20% increase in bus fares would yield less revenue and lower aggregate transportation costs does not mean that inner-city residents would NOT suffer a big loss in utility from the change. They would reallocate their travel modes or change their work and shopping patterns to accommodate the increased fares to a considerable extent. These changes themselves could decrease utility extensively.
• 2. Information on the hypothetical Canadian data set is contained in Exhibit B.
  • a.) As in the U.S. sample, Regression B1 implies that the effect of a one-unit change in price on bus trips is -.50000. In this case, the point estimate is highly insignificant, since the t-ratio of -0.09505 is close to zero under any degrees of freedom scenario. Likewise, the P-value of 0.463 places us solidly in the middle of the t-distribution. Demand, if it has any elasticity at all, appears to be inelastic at the means of the data, since the elasticity point estimate is -0.2106.
  • b.) Regression B2 includes income as an additional control variable. The activist's implicit assertion of a zero effect of price on quantity demanded still cannot be rejected. The t-test statistic for this hypothesis is only -0.6058, still a very plausible value if the null hypothesis is true.
  • c.) In this case, the addition of the previously excluded variable has no appreciable effect on the magnitude of the point estimate of the price effect (it is still right around -0.5). Neither does the expected vast increase in statistical significance materialize. Why? Unlike in the U.S. case, where incomes were strongly correlated with the level of bus fares, in the Canadian data, the correlation between P and Y is only around 0.05. The omitted variable bias in the first regression shrinks the absolute value of the slope on price (but only just a little, instead of hugely as in the U.S. case).
  • d.) Income in the Canadian data appears to have been measured in dollars, rather than in thousands of dollars, since the mean income in the Canadian sample STAT output is $19510. Thus the coefficient on income is much tinier (to compensate). We know from the homework that addressed changes of location and scale in regression variables that if we measured Canadian income in thousands of dollars, the coefficient would be 0.94757, rather than 0.00094757. This would be much more comparable to the U.S. results.
  
  SHORT QUESTIONS:
• 3. Adjusted R-squared values are preferred to regular R-squared values when you are comparing two regressions with different dependent variables? False. R-squared values can ONLY be used to compare goodness-of-fit between regressions with the identical dependent variable, since R-squared is the proportion of variation in the dependent variable explained by the set of X variables being used. Adjusted R-squared values are preferred when comparing regressions using different numbers of explanatory variables, since R-squared alone will certainly increase with additional regressors. Adjusted R-squared invokes a degrees of freedom correction that penalizes models somewhat for using more regressors.
• 4. In ordinary least squares regression, the maintained hypotheses of "homoscedasticity" and "no serial correlation in the errors" do not matter in our derivation of the formulas for b1 and b2. They only come into play as we determine whether these two estimators are unbiased. False. They come into play when we are trying to derive simple formulas for the variances of the slope and intercept estimates. No serial correlation let us ignore the usual covariance terms in the formula for the variance of a linear combination of random variables. With independent observations (a random sample), all the covariances are zero. Homoscedasticity lets us assume that all of the conditional variances of the random Y values were identical (and equal to sigma-squared), so they could be factored out of the summation over i. Then the terms in the squared deviations of the x- variable canceled and we were left with a very tidy expression for the variance of the slope, for example. This was covered in a class handout.
• 5. When we construct a t-test statistic of the null hypothesis that some true underlying population parameter Bj = c (where c is some number), and we find a value for our test statistic of, say, 6.3, what are the two possible conclusions we could draw, which one do we usually draw, and why? One possibility is that we just have a rare value of this t-distributed variable. The probability density function ranges from negative infinity to positive infinity, so very small or very large values are possible, but very unusual. The other possibility is that the null hypothesis that the parameter equals c is incorrect, which could also explain why we have an unusual value. Typically, we assume that if we find a value of the t- variable that would tend to occur less than 5% of the time when the null hypothesis is true, we choose instead to reject the null hypothesis. Of course, 5% of the time, we will be doing this incorrectly, but that's the hazard in statistics.
• 6. What is perfect multicollinearity and why do we require that it not be present if we are going to run a multiple regression? With perfect multicollinearity, one explanatory variable is a perfect linear function of another. They always vary "lock-step" in the data; one never changes without the other changing in a predefined fashion. Thus, we cannot separate out the distinct effects of either one. The "footprint" of the scatter of data over the "floor" or "domain" of the regression function is a straight line. In the three-variable model used in class, we could successfully fit a line through the three-dimensional plot, but not any single plane. To fit the play, we need the two regressor variables to be "spread out".
• BONUS QUESTION: In Regression A1, what is the R- squared value? What is the adjusted R-squared value? Show your work. Since this is a simple regression, the R-squared is the same as the squared value of the pairwise correlation between Q and P, which is thus the square of (-0.41686). Getting the adjusted R-squared without the Analysis of Variance from Means output is tougher. It takes some careful rearranging of the formulas provided on the crib sheet with the exam, but (1 - Adjusted R- squared) equals [(n-1)/(n-k-1)] times (1 - ordinary R-squared). Since you have n=17, k=2, and ordinary R-squared, you have all the ingredients. I would be duly impressed by anybody who had time during the exam to sort this last part out. Well done.