Gasoline Consumption Analysis

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1 Gasoline Consumption Analysis One of the most basic topics in economics is the supply/demand curve. Simply put, the supply offered for sale of a commodity is directly related to its price, while the demand for purchase of a commodity is inversely related to its price. Can this pattern be quantified with real data? (original data source: Economic Report of the President) The following analysis is based on figures for gasoline consumption in the United States from 1960 through First, here is a scatter plot of gasoline consumption per capita versus the price index for gasoline. The reason we take log to the prices is that, according to economic theory, coefficients from a log/log fit can be viewed as elasticity Scatterplot of loggpc vs logpg 1.05 loggpc logpg There is apparently a positive relationship here, not an inverse one! Is economic theory all wrong? Well, no. We haven t taken into account the fact that potential determinants of gasoline consumption other than price have also changed over the years. We can do this using multiple regression. Variables that are used are as follows: logpg : Logged price index for gasoline logi : Logged per capita real disposable income logpnc : Logged price index for new cars logpuc : Logged price index for used cars logppt : Logged price index for public transportation logpd : Logged aggregate price index for durable goods logpn : Logged aggregate price index for nondurable goods logps : Logged aggregate price index for consumer services

2 First, we notice that these variables are highly correlated with each other. A regression using all of the variables will exhibit high multicollinearity, and will presumably include several redundant variables. Correlations: logpg, logi, logpnc, logpuc, logppt, logpd, logpn, logps logpg logi logpnc logpuc logppt logpd logpn logi logpnc logpuc logppt logpd logpn logps Here is the regression output. Just as we suspected, we have several redundant variables (ones with the t statistics being insignificant). Note the extremely high VIF values, by the way. Regression Analysis: loggpc versus logpg, logi,... loggpc = logpg logi logpnc logpuc logppt logpd logpn logps Constant logpg logi logpnc logpuc logppt logpd logpn logps S = R-Sq = 98.9% R-Sq(adj) = 98.6% This leads us to the general problem of model selection. Omitting important effects (underfitting) reduces predictive power, biases estimates of effects for included predictors, and results in less understanding of the process being studied. Including unnecessary effects (overfitting) complicates descriptions of the process, and tends to lead to poorer predictions because of the additional unnecessary noise. Model selection is as much of an art as a science, and involves subjective decision-making. We have already seen how hypothesis tests (e.g. t-tests) can be used to try to decide which predictors are needed in a model. Unfortunately, there are several reasons why such tests are not adequate for the task of choosing the appropriate model to use. First, hypothesis tests don t necessarily answer the question a data analyst is most interested in. With a large enough sample, almost any estimated slope will be significantly different from zero, but that doesn t mean that the predictor provides additional useful predictive power. Similarly, in small samples, important effects might not be statistically significant at typical levels simply because of insufficient data. That is, there is a clear distinction between statistical significance and practical importance.

3 A second important point is that when predictors are related to one another (as is the case for these data), t-test can provide very misleading indications of the importance of a predictor. For example, consider a two-predictor situation where the predictors are each highly correlated with the target variable, and are also highly correlated with each other. In this situation it is likely that the t-statistic for each predictor will be relatively small. This is not an inappropriate result, as t-statistics test whether a predictor adds significant predictive power given the other variables in the model. For each predictor, given the other, the predictor adds little of importance (being highly correlated with each other, one is redundant in the presence of the other). Unfortunately this means that the t-statistics are useless in identifying that either predictor alone provides great predictive power. What is needed is a strategy for determining a best model (or even better, a set of best models) among a larger class of candidate models. In doing this, we need to be clear as to what we mean by a best model. The underlying premise is that any model is viewed as an approximation of reality. In this sense, there is no true model at all (or, perhaps, what is basically the same, the true model is too complex to be useful). Our goal is not to find the true model, but rather to find a model, or set of models, that best balances fit and simplicity (the so-called principle of parsimony, using as few parameters as possible while still adequately representing the relationships in the data). This should result in a model that provides useful descriptions of the process being studied from estimated parameters, and which can be used for predictions of future events. An effective way to do this is using a best subsets regression. Here is the output: Best Subsets Regression: loggpc versus logpg, logi,... Response is loggpc l l l l o o o l l l o l g g g o o o g o P P P g g g Mallows P g N U P P P P Vars R-Sq R-Sq(adj) C-p S G I C C T D N S X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Since we are interested in estimating a demand function, we would like to consider models with the gasoline price variable logpg included. We would like to find the number of predictors where adding any more doesn t improve the fit very much; that is, where the R 2 levels off. One way to make this decision is to try to find a place where the standard error of the estimate begins to level off (since at that point the width of prediction intervals won t change very much if more predictors are used). That appears to be at the five predictor model. The adjusted R 2, on the other hand, is maximized at the seven predictor model (note that the adjusted R 2 attempts to

4 explicitly represent the tradeoff between fit and complexity of the model by penalizing R 2 with a term that is a function of the number of predictors p). The entry under C-p refers to Mallows Cp, a criterion introduced in the late 1960 s that was devised to explicitly balance the desires for improved fit and simplicity of the model by being an estimate of the error that would come from using the model for prediction. Cp has the form Cp = SSE/s 2 n + 2p + 2, where SSE is the Residual SS for the model being examined, p is the number of predictors in that model, and s 2 is the Residual MS based on using all of the candidate predicting variables. Minimization of Cp forces a tradeoff between fidelity to the data (smaller Residual SS) and simplicity of the model (smaller p). Two operational rules for Cp that have been suggested are to choose the model with smallest Cp, or smallest p such that Cp = p+1 or less. Both rules recommend the seven variable model. For this dataset, the different methods agree that the models to consider have somewhere between five and seven predictors. Seven predictors seems too many here (given the gain in R 2 of 0.1% when going from six to seven predictors), and the t statistic for logppt in this model is not significantly different from zero (logpuc is marginal): Regression Analysis: loggpc versus logpg, logi,... loggpc = logpg logi logpuc logppt logpd logpn logps Constant logpg logi logpuc logppt logpd logpn logps S = R-Sq = 98.9% R-Sq(adj) = 98.7% The best six predictor model is the same as above, except that the insignificant logppt is dropped out. The t statistic for logpuc is still marginal, but by this output this model is not unreasonable: Regression Analysis: loggpc versus logpg, logi,... loggpc = logpg logi logpuc logpd logpn logps Constant logpg logi logpuc logpd logpn logps S = R-Sq = 98.8% R-Sq(adj) = 98.6%

5 Ultimately, when you have competing models that are this similar, it becomes a matter of personal preference. We could also drop logpuc and look at the resulting five predictor model. Regression Analysis: loggpc versus logpg, logi, logpd, logpn, logps loggpc = logpg logi logpd logpn logps Constant logpg logi logpd logpn logps S = R-Sq = 98.6% R-Sq(adj) = 98.4% All of the variables are significant, but also show a lot of collinearity. The key thing to recognize from this model is that the coefficient for the price of gasoline (logpg) in the equation is Recall what this means: given that income, durable goods price index, nondurable goods price index, and consumer services price index are held fixed, a one unit increase in the logged gasoline price index is associated with a.4985 unit decrease in logged gasoline consumption. This, of course, is the inverse relationship we are looking for in a demand curve. As noted earlier, in this log/log model, this value represents the estimated price elasticity of gasoline, saying that a 1% increase in price is associated with a 0.5% decrease in consumption. The high variance inflation factors lead to a larger standard error of the coefficient for logpg. In fact, a 95% confidence interval for the decrease in logged gasoline consumption associated with a one unit increase in the logged price index is ( , ). There is an important issue regarding model selection that cannot be ignored. The act of looking at all of the regressions (as best subsets does) will cause an inflation of measures of the strength of the regression, since apparent relationships that are due to random chance will be identified as a true relationship. For example, in a sample of 400 observations with 50 potential predictors that are pure noise, best subsets regression is very likely to identify a model with R 2 at least 50%! This problem is sometimes given the names of data snooping. For this reason, it is not a good idea to just throw every possible variable in, and then let the computer choose the ones to use. Rather, there should be a very thorough pre examination of the data, where irrelevant and redundant variables are weeded out before any statistics are run. Even better, if you have enough data, some should be held out of the analysis; then, the model built on the first part of the sample can be validated on the held out portion. If it s not possible to validate your model on new data, a simple adjustment to the standard error of the estimate can help. The value reported above,.00828, is based on the number of predictors in the chosen model (that is, 5). A better choice is to use the number of predictors in the most complex model considered in the model selection, or in this case 10. So, an adjusted version of the standard error of the estimate divides by , rather than , or sqrt( /25) = Thus, a better rough 95% prediction interval for logged per capita demand when using this model is ±(2)(.00907) = ±.0181, rather than ±(2)(.00828) = ±.0166, with the former roughly 9% wider than the latter. Diagnostics and residual plots look okay here.

6 Finally, we can try to make an adjusted demand versus gasoline price index plot. That is, the relationship between demand for gasoline and price after all of the other variables have been taken into account. Here s how we do it: (1) Do a regression of the dependent variable (here loggpc) on all of the predictors except the one you re interested in (here logpg). Save the residuals (call it RES1). We can call this variable the adjusted gasoline consumption. (2) Do a regression of the predicting variable you re interested in (here logpg) on the other predicting variables. Save the residuals (call it RES2). We can call this variable the adjusted gasoline price index. (3) The partial correlation is simply the correlation between RES1 and RES2. (4) If you regressed RES1 on RES2 you d notice something remarkable the coefficient for RES2 is precisely the same as the coefficient for the predictor of interest in the full multiple regression model (that is, the coefficient for RES2 in this case is exactly ). (5) If you d like a graphical representation of the adjusted relationship between the predictor of interest and the target variable, taking the other predictors into account, just do a scatter plot of RES1 on RES2. Thus, here is the (adjusted) demand curve for gasoline: 0.03 Scatterplot of RES1 vs RES RES RES

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