Economics 300. Econometrics Multiple Regression: Extensions and Issues

Transcription

1 Economics 300 Econometrics Multiple : Extensions and Dennis C. Plott University of Illinois at Chicago Department of Economics Fall 2014 Dennis C. Plott (UIC) ECON 300 Fall / 36

2 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

3 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

4 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

5 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

6 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u y = β 0 + β 2 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

7 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u y = β 0 + β 2 1 x β k x k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

8 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u y = β 0 + β 2 1 x β k x k + u y = β 0 + β 1 x β kx k + u 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

9 The Classical s The Classical s The classical assumptions must be met in order for OLS estimators to be the best (i.e., minimum variance) linear unbiased estimators (BLUE) The seven 1 classical assumptions are: 1. The population model is linear in parameters (i.e., β) and correctly specified. y = β 0 + β 1 x β k x k + u y = β 0 β 3 + β 1 x β k x k + u y = β 0 + β 2 1 x β k x k + u y = β 0 + β 1 x β kx k + u 2. {x i,y i } are a random sample; i.e., each individual in a population is equally likely to be picked. Also, all data points come from the same population. 1 five to seven, depends on source Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

10 The Classical s The Classical s (Continued) 3. None of the independent variables is constant and there are no exact linear relationships; no perfect collinearity in the regressors. y = β 0 + β 1 x 1 + β 2 x 2 + u x 1 = γ 0 + γ 1 x 2 i.e., if I know x 2, then I know x 1. For example, housep = β 0 + β 1 sqft + β 2 sqm + u. 4. Zero conditional mean of errors! E(u i x i ) = E(u i ) = 0 If, for example, I regress wage on educ and I know that someone has 16 years of education, then this does not help me to predict if they are above, below, or on the population regression line. Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

11 The Classical s The Classical s (Continued) 5. Homoskedastic Errors; i.e., constant variance 6. No Serial Correlation Var(u i x i ) = σ 2 = constant, i Cov(u i,u j ) = 0, i j This means errors are independent knowing one of the errors, does not help predict another error. 7. (Optional) Normally Distributed Errors u N(0,σ 2 u ) This also implies that the conditional distribution of y is normally distributed The normality assumption does not apply to the independent variables Dennis C. Plott (UIC) The Classical s ECON 300 Fall / 36

12 Symptoms Irrelevant See do-file. The t-statistics for the coefficients are not significant. Yet, the overall F-statistic is significant. Even though both independent variables have the same standard errors and almost identical correlations with y, their estimated effects are radically different. x 1 and x 2 are very highly correlated. The sample is relatively small. These are all indicators that multicollinearity might be a problem in these data. Dennis C. Plott (UIC) ECON 300 Fall / 36

13 Irrelevant Causes Improper use of dummy variables; e.g., failure to exclude one category. Including a variable that is computed from other variables in the equation (e.g., family income = husband s income + wife s income, and the regression includes all three income measures). In effect, including the same or almost the same variable twice (height in feet and height in inches.) The above all imply some sort of error on the researcher s part. Sometimes models include variables that are nonlinear or non-additive (interactive) functions of other variables in the model. For example, the independent variables might include both x and x 2. In such cases, the original variables and the variables computed from them can be highly correlated. Also, multicollinearity between nonlinear and non-additive terms could make it difficult to determine whether there is multicollinearity involving other variables. Dennis C. Plott (UIC) ECON 300 Fall / 36

14 Irrelevant Consequences Even extreme multicollinearity (so long as it is not perfect) does not violate OLS assumptions. OLS estimates are still unbiased and BLUE. Nevertheless, the greater the multicollinearity, the greater the standard errors. When high multicollinearity is present, confidence intervals for coefficients tend to be very wide and t-statistics tend to be very small. Coefficients will have to be larger in order to be statistically significant, i.e. it will be harder to reject the null when multicollinearity is present. Note, however, that large standard errors can be caused by things besides multicollinearity. When two independent variables are highly and positively correlated, their slope coefficient estimators will tend to be highly and negatively correlated. Further, a different sample will likely produce the opposite result. In other words, if you overestimate the effect of one parameter, you will tend to underestimate the effect of the other. Hence, coefficient estimates tend to be very shaky from one sample to the next. Dennis C. Plott (UIC) ECON 300 Fall / 36

15 Detection & Solution Irrelevant Detection is a matter of degree. There is no irrefutable test that it is or is not a problem. See symptoms; slide 5. In Stata, type vif after a regression. Solution The Variance Inflation Factor (VIF) shows us how much the variance of the coefficient estimate is being inflated by multicollinearity. For example, if the VIF for a variable were 9, its standard error would be three times as large as it would be if its VIF was 1. In such a case, the coefficient would have to be 3 times as large to be statistically significant. More and better data. Dennis C. Plott (UIC) ECON 300 Fall / 36

16 Irrelevant Recall homoskedasticity: Var(u i x i ) = σ 2 = constant, i. Notice the lack of a subscript on the sigma. If the error terms do not have constant variance, they are said to be heteroskedastic: Var(u i x i ) = σ 2 constant. Notice the subscript i on the sigma; i i.e., the variance will depend on observation of the independent variable. Non Sequitur: heteroskedastic comes from Ancient Greek with hetero (different) and skedasis (dispersion). Dennis C. Plott (UIC) ECON 300 Fall / 36

17 Irrelevant Occurrence Errors may increase as the value of an independent variable increases. For example, consider a model in which annual family income is the independent variables and annual family expenditures on vacations is the dependent variable. Families with low incomes will spend relatively little on vacations, and the variations in expenditures across such families will be small. But for families with large incomes, the amount of discretionary income will be higher. The mean amount spent on vacations will be higher, and there will also be greater variability among such families, resulting in heteroskedasticity. Note that, in this example, a high family income is a necessary but not sufficient condition for large vacation expenditures. Any time a high value for an independent variable is a necessary but not sufficient condition for an observation to have a high value on a dependent variable, heteroskedasticity is likely. Errors may also increase as the values of an independent variable becomes more extreme in either direction. For example, with attitudes that range from extremely negative to extremely positive. This will produce something that looks like an hourglass shape or bow-tie. Dennis C. Plott (UIC) ECON 300 Fall / 36

18 Irrelevant Occurrence (Continued) Measurement error can cause heteroskedasticity. Some respondents might provide more accurate responses than others. (Note: this problem arises from the violation of another assumption: variables are measured without error; i.e., zero conditional mean.) can also occur if there are sub-population differences or other interaction effects; e.g., the effect of income on expenditures differs for whites and blacks. (Again, the problem arises from violation of the assumption that no such differences exist or have already been incorporated into the model; i.e., the model is correctly specified.) Other model misspecifications can produce heteroskedasticity. For example, it may be that instead of using y, you should be using the log of y. Instead of using x, maybe you should be using x 2, or both x and x 2. Important variables may be omitted from the model. If the model were correctly specified, you might find that the patterns of heteroskedasticity disappear. Dennis C. Plott (UIC) ECON 300 Fall / 36

19 Consequences Irrelevant Note that heteroskedasticity is often a by-product of other violations of assumptions. These violations have their own consequences which we will deal with elsewhere. For now, we will assume that other assumptions except heteroskedasticity have been met. Then, does NOT result in biased parameter estimates. However, OLS estimates are no longer BLUE. That is, among all the unbiased estimators, OLS does not provide the estimate with the smallest variance. Depending on the nature of the heteroskedasticity, significance tests can be too high or too low. In addition, the standard errors are biased when heteroskedasticity is present. This in turn leads to bias in test statistics (e.g., t- and F-statistics) and confidence intervals. Dennis C. Plott (UIC) ECON 300 Fall / 36

20 Irrelevant Detection Visual Inspection Do a visual inspection of residuals plotted against fitted values; or, plot the independent variable suspected to be correlated with the variance of the error term. In Stata, after running a regression, you could use the rvfplot (residuals versus fitted values) or rvpplot command (residual versus predictor plot; e.g., plot the residuals versus one of the x variables included in the equation). Formal Tests (A bunch, differ in the form of heteroskedasticity tested and whether or not the error is Normally distributed.) Breusch-Pagan/Cook-Weisberg Test for designed to detect any linear form of heteroskedasticity. The null hypothesis is that the error variances are all equal versus the alternative that the error variances are a multiplicative function of one or more variables. In Stata, after running a regression, use the following command: estat hettest. White s General Test for Data Type In Stata, estat imtest, white. occurs most often in cross-sectional data. These are data where observations are all for the same time period (e.g., a particular month, day, or year) but are from different entities (e.g., people, firms, provinces, countries, etc.) Dennis C. Plott (UIC) ECON 300 Fall / 36

21 Irrelevant Solution Respecify the /Transform the As noted before, sometimes heteroskedasticity results from improper model specification. Note: specification is determined from theory. Research and think carefully about what belongs in the model prior to running regressions. There may be subgroup differences. Effects of variables may not be linear. Perhaps some important variables have been left out of the model. Use Robust Standard Errors or Cluster Stata includes options with most routines for estimating robust standard errors (you will also hear these referred to as Huber/White estimators or sandwich estimators of variance). As noted above, heteroskedasticity causes standard errors to be biased. OLS assumes that errors are both independent and identically distributed; robust standard errors relax either or both of those assumptions. Hence, when heteroskedasticity is present, robust standard errors tend to be more trustworthy. Dennis C. Plott (UIC) ECON 300 Fall / 36

22 Irrelevant Solution (Continued) The use of robust standard errors does not change coefficient estimates, but (because the standard errors are changed) the test statistics will give you reasonably accurate p-values. With Stata, robust standard errors can usually be computed via the addition of two parameters, robust and cluster. The robust option relaxes the assumption that the errors are identically distributed, while cluster relaxes the assumption that the error terms are independent of each other. Caution: Do not confuse robust standard errors with robust regression. Despite their similar names, they deal with different problems: Robust standard errors address the problem of errors that are not independent and identically distributed. The use of robust standard errors will not change the coefficient estimates provided by OLS, but they will change the standard errors and significance tests. Dennis C. Plott (UIC) ECON 300 Fall / 36

23 Irrelevant Irrelevant This refers to the case of including a variable in an equation when it does not belong there. In other words, not derived from (sound) economic theory; i.e., corr(x 1,x 2 ) 0 Assume that the true regression specification is: y i = β 0 + β 1 x 1 + u (1) But the researcher for some reason includes an extra variable: y i = β 0 + β 1 x 1 + β 2 x 2 + v (2) The misspecified equation s error term then becomes: v = u β 2 x 2 (3) Dennis C. Plott (UIC) Irrelevant ECON 300 Fall / 36

24 Irrelevant (Continued) Irrelevant So, the inclusion of an irrelevant variable will not cause bias (since the true coefficient of the irrelevant variable is zero, and so the second term will drop out of Equation 3) However, the inclusion of an irrelevant variable will: Increase the variance of the estimated coefficients, and this increased variance will tend to decrease the absolute magnitude of their t-scores. Decrease the R 2, but not the R 2 Dennis C. Plott (UIC) Irrelevant ECON 300 Fall / 36

25 Internal Validity Irrelevant Internal validity: the statistical inferences about causal effects are valid for the population being studied Five threats to internal validity 1. Omitted variable bias Wrong functional form 2. Sample selection bias 3. Simultaneous causality bias (or reversed causality) 4. Measurement error ( Errors-in-variables bias) All threats lead to a violation of the conditional independence assumption (zero conditional mean assumption) Dennis C. Plott (UIC) ECON 300 Fall / 36

26 Omitted Variable Bias Irrelevant Two reasons why an important explanatory variable might have been left out: 1. we forgot it is not available in the dataset, we are examining Either way, this may lead to omitted variable bias (or, more generally, specification bias) The reason for this is that when a variable is not included, it cannot be held constant Omitting a relevant variable usually is evidence that the entire equation is a suspect, because of the likely bias of the coefficients. Dennis C. Plott (UIC) ECON 300 Fall / 36

27 Irrelevant Omitted Variable Bias Consequences Suppose the true regression model is: y i = β 0 + β 1 x 1 + β 2 x 2 + u (4) where u is a classical error term and corr(x 1,x 2 ) 0 If x 2 is omitted, the equation becomes instead: y i = β 0 + β 1 x 1 + v (5) Where: v = u + β 2 x 2 (6) Hence, the explanatory variables in the estimated regression (5) are not independent of the error term (unless the omitted variable is uncorrelated with all the included variables) Dennis C. Plott (UIC) ECON 300 Fall / 36

28 Irrelevant Omitted Variable Bias Consequences (Continued) What happens if we estimate Equation (5) when Equation (4) is correct? Bias This means: E( ˆβ 1 ) β 1 (7) The amount of bias is a function of the impact of the omitted variable on the dependent variable times a function of the correlation between the included and the omitted variable So, the bias exists unless: 1. the true coefficient equals zero, or 2. the included and omitted variables are uncorrelated Dennis C. Plott (UIC) ECON 300 Fall / 36

29 Omitted Variable Bias Potential Solutions Irrelevant include variable if measurable (or proxy) use panel data (solves problem when there are time-constant individual effects) use instrumental variable regression run a randomized experiment Dennis C. Plott (UIC) ECON 300 Fall / 36

30 Irrelevant Endogeneity and Unobserved Heterogeneity Omitted variable bias is the most common illustration of what economists refer to as endogeneity. Endogenous variables are variables determined by other variables in the system, while exogenous variables are variables which can be considered external shocks to the system. Other important sources of endogeneity include reverse causality and measurement error. Functional misspecification is really a special case of omitted variable bias. To truly be able to make a causal claim, we need a truly exogenous variable that is, a variable which is not related to any of the other variables in the system, unobserved and observed. The problem with observational data is that there are an infinite number of unobserved variables which could render our observed relationship endogenous. This is the problem of unobserved heterogeneity in our sample. Dennis C. Plott (UIC) ECON 300 Fall / 36

31 Endogeneity and Unobserved Heterogeneity (Continued) Irrelevant As an example, let s look at a simple question. Do private schools improve student s test performance? Let s say we had a sample of public and private school students math test scores. We could look at the difference in the average score between groups. But it would be dangerous to assume that such a difference reflected the "treatment" of private schools, because it seems likely that more apt students are more likely to self-select into private schools. One standard solution is to control for all the observed measures that might lead to such self-selection which are available to us. The problem is that we are unlikely to effectively control for all of this selectivity, because some variables associated with the selection process are probably unobserved. Even if all the important variables were observed, we would only completely control for them if we correctly specified the functional form of their relationship to test scores. Dennis C. Plott (UIC) ECON 300 Fall / 36

32 Irrelevant Measurement Error (Errors in ) Many economists feel that the greatest drawback to econometrics is the fact that the data with which econometricians must work are so poor. A well-known quotation expressing this feeling is due to Josiah Stamp: The Government are very keen on amassing statistics they collect them, add them, raise them to the nth power, take the cube root and prepare wonderful diagrams. But what you must never forget is that every one of those figures comes in the first instance from the village watchman, who just puts down what he damn pleases. The errors-in-variables problem is concerned with the implication of using incorrectly measured variables, whether these measurement errors arise from the whims of the village watchman or from the use by econometricians of a proxy variable in place of an unobservable variable suggested by economic theory. Examples survey data on households: recall bias: how much time did you spend unemployed last year? rounding bias: how much money did you spend on food last week? Dennis C. Plott (UIC) ECON 300 Fall / 36

33 Irrelevant Measurement Error (Continued) Errors in measuring the dependent variables are incorporated in the disturbance term; their existence causes no problems. provided the measurement error in y is uncorrelated with (correctly measured) x When there are errors in measuring an independent variable, however, the zero conditional mean assumption is violated, since these measurement errors make this independent variable stochastic; the seriousness of this depends on whether or not this regressor is distributed independently of the disturbance. If only one of the explanatory variables is measured with error, it can be shown: the OLS estimator of the coefficient on that variable is biased towards zero (attenuation bias) this bias does NOT disappear in large samples (OLS is inconsistent) the OLS estimator of the coefficients on the other explanatory variables are also biased, in unknown directions If several explanatory variables are measured with error, it is very difficult to sign the biases for any of the coefficients Dennis C. Plott (UIC) ECON 300 Fall / 36

34 Irrelevant Measurement Error (Continued) Morgenstern (1963) wrote an entire book examining the accuracy of economic data. Some spectacular examples of data fudging by government agencies can be found in Streissler (1970, pp. 27 9). Example: a large overstatement of housing starts in Austria was compensated for by deliberately understating several subsequent housing start figures. Streissler claims that often the econometrician more or less completely misunderstands what the statistics he works with really mean. Griliches (1985) offers four responses to Morgenstern: 1. The data are not that bad. 2. The data are lousy but it doesn t matter. 3. The data are bad but we have learned how to live with them and adjust for their foibles. 4. That is all there is it is the only game in town and we have to make the best of it. In some instances it could be argued that economic agents respond to the measured rather than the true variables, implying that the original estimating equation should be specified in terms of the measured rather than the true values of the regressors. This eliminates the errors-in-variables problem. Dennis C. Plott (UIC) ECON 300 Fall / 36

35 Irrelevant Sample Selection Bias Often assumed in regression framework: simple random sampling of the population, but sometimes sample selects itself Sample selection bias arises when a selection process Influences the availability of data and that process is related to the dependent variable. Induces correlation between regressor and error term (again violates conditional mean independence) Example: Returns to Education Random sample from the population of workers, data on earnings and years of education Problem: factors that determine whether someone works are quite similar to the factors that determine how much that person earns when employed Potential Solutions Randomized controlled experiment Construct a model of the sample selection problem and estimate that model (Heckman s sample selection model (1979) which was cited in his 2000 Nobel) Dennis C. Plott (UIC) ECON 300 Fall / 36

36 Simultaneity (Reverse Causality) Irrelevant Example of reversed causality: low wages cause bad health or bad health causes low wages? Potential solutions: Randomized controlled experiment: because x is chosen at random by the experimenter, there is no feedback from the outcome variable to y (assuming perfect compliance) Use instrumental variables regression to estimate the causal effect of interest (use the variation in x that is exogenous) Develop and estimate a complete model of both directions of causality (idea behind large structural models, very difficult in practice). Dennis C. Plott (UIC) ECON 300 Fall / 36

37 Binary Linear Probability (LPM) Dummy In this class of models, we consider the case where the dependent variable can take the value of 0 or 1. They are often termed dichotomous variables. The literature on this type of model is extensive, it can include cases where there are more than 2 possible outcomes, however we are only covering an introductory section of this area of econometrics. These types of model tend to be associated with the cross-sectional econometrics rather than time series. There are many examples of this type of model in the finance literature. For instance, if we were to examine bank failures. { 1,if the bank fails y = 0, otherwise Dennis C. Plott (UIC) Binary ECON 300 Fall / 36

38 Data Binary Linear Probability (LPM) When examining the dummy dependent variables we need to ensure there are sufficient numbers of 0s and 1s. If we were assessing bank failures, we would need a sample of both banks that have failed and those that have not failed. This can create problems with sampling, as it is easier to find data for solvent banks relative to failed banks. Dennis C. Plott (UIC) Binary ECON 300 Fall / 36

39 Linear Probability (LPM) Binary Linear Probability (LPM) The Linear Probability, uses OLS to estimate the model, the coefficients, t-statistics, etc. are then interpreted in the usual way. This produces the usual linear regression line, which is fitted through the two sets of observations. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

40 Features of the LPM Binary Linear Probability (LPM) The dependent variable has two values, the value 1 has a probability of p and the value 0 has a probability of (1 p). This is known as the Bernoulli probability distribution. In this case the expected value of a random variable following a Bernoulli distribution is the probability the variable equals 1. Since the probability of p must lie between 0 and 1, then the expected value of the dependent variable must also lie between 0 and 1. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

41 Binary Linear Probability (LPM) Problems with LPM The error term is not normally distributed, it also follows the Bernoulli distribution. The variance of the error term is heteroskedastistic. The variance for the Bernoulli distribution is p(1 p), where p is the probability of a success. Possibly the most problematic aspect of the LPM is the non-fulfillment of the requirement that the estimated value of the dependent variable y lies between 0 and 1. One way around the problem is to assume that all values below 0 and above 1 are actually 0 or 1 respectively. An alternative remedy to the problem is to use a technique such as the Logit or Probit models. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

42 Problems with LPM (Continued) Binary Linear Probability (LPM) The final problem with the LPM is that it is a linear model and assumes that the probability of the dependent variable equaling 1 is linearly related to the explanatory variable. For example if we have a model where the dependent variable takes the value of 1 if a mortgage is granted to a bank customer and 0 otherwise, regressed on the customers income. The probability of being granted a mortgage will rise steadily at low income levels, but change hardly at all at high income levels. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

43 Binary Linear Probability (LPM) LPM Example with Interpretation highbp i = age i lincome i ( se) ˆ (0.124) (0.001) (0.076) { 1,if have high blood pressure highbp = 0, otherwise (0.021) diabetes i Interpretation Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

44 Binary Linear Probability (LPM) LPM Example with Interpretation highbp i = age i lincome i ( se) ˆ (0.124) (0.001) (0.076) { 1,if have high blood pressure highbp = 0, otherwise (0.021) diabetes i Interpretation age: On average, a one year increase in age will increase the probability of having high blood pressure by 0.5 percentage points, ceteris paribus. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

45 Binary Linear Probability (LPM) LPM Example with Interpretation highbp i = age i lincome i ( se) ˆ (0.124) (0.001) (0.076) { 1,if have high blood pressure highbp = 0, otherwise (0.021) diabetes i Interpretation age: On average, a one year increase in age will increase the probability of having high blood pressure by 0.5 percentage points, ceteris paribus. lincome: On average, a 100% increase in income will reduced the probability of having high blood pressure by 4.1 percentage points, ceteris paribus. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36

46 Binary Linear Probability (LPM) LPM Example with Interpretation highbp i = age i lincome i ( se) ˆ (0.124) (0.001) (0.076) { 1,if have high blood pressure highbp = 0, otherwise (0.021) diabetes i Interpretation age: On average, a one year increase in age will increase the probability of having high blood pressure by 0.5 percentage points, ceteris paribus. lincome: On average, a 100% increase in income will reduced the probability of having high blood pressure by 4.1 percentage points, ceteris paribus. diabetes: On average, people with diabetes have 19.2 percentage point higher probability of having high blood pressure than those without diabetes, ceteris paribus. Dennis C. Plott (UIC) Linear Probability (LPM) ECON 300 Fall / 36