Basics of Regression Analysis

# Basics of Regression Analysis - Basics of Regression...

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Unformatted text preview: Basics of Regression Analysis Regression Analysis is aiming at quantifying a functional relationship between a dependent variable (y) and one ore more independent variables (x). For instance, assume the height of the son is a function of his father’s height, H S = f ( H F ) and we draw on 730 observations (see Table). A regression estimates the line of best fit through the scatter plot shown below. Technically, an “ordinary least square” or OLS ­regression minimizes the sum of the squared deviations (also called residuals) ! from the fitted line. Data Format We distinguish between (a) Time Series data (b) Cross Section data and (c) Pooled data (combined cross section and time series) The model above analyses the height of several people at one point in time and is thus a cross sectional analysis. We assume a statistical model of the form H S i = b0 + b1H Fi + "i where i stands for each observation and " denotes a randomly ! ! distributed error term, i.e., the deviation from the estimated height (i.e., the residuals). ! Regressions can have more than one independent variable (multi ­variate regression). For instance, the quantity consumed of a certain good QG can be a function of the goods price PG, the price of a substitute PS and the consumer’s income YD): QG = f ( PG , PS ,YD ) Important Features: (1) Ground in Theory Selecting the independent variable should ground in (economic) theory. (2) The sign of the coefficient Make sure that the sign of the coefficient squares with your expectations. (3) R2 Check the R2, which is the goodness ­to ­fit. The following three Figures depict R2s of 0, 0.95 and 1. Since, in this example, we have 3 independent variables, we call There is no critical value of a “sufficient R2”. But in general, we are happy with an R2>0.5 for time series analyses and R2>0.3 in cross sections. (4) T ­Statistics The t ­statistic provides you with the significance level of the estimated coefficients. Technically, the t ­statistics is calculated as the value of the coefficient divided by the standard error. If the t ­statistics is insignificantly low (that is, the standard errors are high) we cannot trust the estimated coefficient (i.e., we cannot rule out that the “true” coefficient is zero). Normally, we do not want the likelihood this to happen to be greater than 5%. For the t ­statistics, this translates into a minimum absolute value of 2 (this is just a rule of thumb). (5) Marginal effects and elasticities We distinguish three forms of specifications, linear, log ­linear and log ­log equations.  ­ Linear Equations take on the form y = bo + b1 x + ! . Here the marginal effect of a one ­unit change in x on y is equal to b1. Example: y = 0.035*x  a one unit increase in x leads to a 0.035 unit increase in y.  ­ Log ­Linear Equations take on the form ln y = bo + b1 x + ! . Here the marginal effect of a one ­unit change in x is equal to a b1 ­precentage change in y. Example: ln(y) = 0.035*x  a one unit increase in x leads to a 3.5% increase in y.  ­ Log_Log Equations take on the form ln y = bo + b1 ln x + ! . Here the marginal effect of a one ­unit change in x is equal to a b1 ­precentage change in y. Thus, the coefficients of log ­log models yield constant elasticities. Example: ln(y) = 0.035*ln(x)  a one ­percent increase in x leads to a 3.5% increase in y. This is identical with a constant elasticity of 0.035. (6) Short ­Run vs. Long ­Run Elasticities Assume a log ­log equations includes a lagged dependent variable (last period’s y), for instance, to model slow reactions of capital stocks. This equation will be given as ln yt = bo + b1 ln yt!1 + b2 ln xt + !t The short ­run elasticity of x is equal to b2. The long ­run elasticity of x is equal to b2/(1 ­b1) Example: ln y = 120 + 0.80 * yt!1 + 0.20 * xt Here the short ­run elasticity of x equals 0.2. The long ­run elasticity of x equals 0.2/(1 ­0.8)=1 ...
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## This note was uploaded on 04/05/2012 for the course ECON UA.31 taught by Professor Storchmann during the Spring '11 term at NYU.

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