Econometrics I. by Kefyalew Endale (AAU)

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1 Econometrics I By Kefyalew Endale, Assistant Professor, Department of Economics, Addis Ababa University ekefyalew@gmail.com October 2016 Main reference-wooldrigde (2004). Introductory Econometrics, A modern Approach, 4 th edition by Kefyalew Endale (AAU) 1

2 Chapter 2: The Simple regression model This model can be used to study the relationship between two variables (Y and X). Examples Y X 1. Cumulative GPA Number hours spent on studying 2. Crop yield (kg/ha) Fertilizer used in kg/ha 3. Wage Years of schooling Three issues in formulating the regression of Y on X How do we allow for the other factors to affect Y? What is the functional relationship between Y and X? How can we be sure we are capturing the Cetrius paribus r/p between Y and X? by Kefyalew Endale (AAU) 2

3 The Simple Linear Regression Model Specifies the relationship between Y and X as a linear relationship Y = β 0 + β 1 X + ε.2.1 Equation (2.1) is called the two variable linear regression model or bivariate regression model β 1 -is the slope (which measures the impact of a unit change in X on Y) β 0 -is the intercept which captures the value of Y when X takes a zero value. Intercept doesn t show relationships between X and Y but it is used in predicting the values of Y for a given values of X by Kefyalew Endale (AAU) 3

4 Terminologies in Simple Linear Regression Model The most frequently used terminologies for Y and X are dependent and Independent variables, respectively. But there are other terms for Y and X Explained variable Response variable Predicted variable Regressand variable Y Explanatory variable Control variable Predictor variable Regressor variable ε-is the error term or disturbance in the relationship which represents factors other than X that affect Y In the simple regression model such as the one in equation (2.1) all factors other than X which affect Y are considered as unobserved factors X by Kefyalew Endale (AAU) 4

5 Simple Linear regression CONT D The change in the dependent variable is given by Y = β 0 + β 1 X + ε..2.2 Cetrius paribus implies that all factors other than X are kept constant ( ε=0 in the case of simple regression). If there is another explanatory variable (Z) in addition to X, then cetrius paribus assumption implies that Z=0 and ε = 0 Then, the marginal impact of X on Y is given by Δy Δx = β 1 Interpretation, as X changes by 1 unit, Y changes by β 1 unit. by Kefyalew Endale (AAU) 5

6 Simple Linear regression CONT D Eg. Wheat Yield (kg/ha)= Fertilizer (kg/ha)+ ε t-ratio (10.06) (6.72) β 0 =1069. It means when the quantity of fertilizer is 0, the wheat yield per hectare of land is 1069kg β 1 = 3.93 slope. It means as the farmer increases the quantity of fertilizer per hectare of land by 1 kg, the wheat yield increases by 3.93 Kg. Questions 1-by how many kilograms will the wheat yield increase if the farmer raises the quantity of fertilizer from 20 (kg/ha) to 25(kg/ha)? 2. What is the predicted value of the wheat yield when fertilizer is 25 kg/ha 3. Is the coefficient of fertilizer statistically significant? Why? by Kefyalew Endale (AAU) 6

7 Simple Linear regression CONT D Limitation of Simple Linear regression-it assumes that the impacts are linear. Raising fertilizer from 5kg/ha to 6kg/ha could have larger marginal effect compared to raising it from 120kg/ha to 121kg/ha Eg2. wage = β 0 + β 1 Years of schooling + ε This model assumes linear impact of years of schooling on wages. But in practice years of schooling have non-linear impacts For instance, increasing years of schooling from grade 7 to grade 8 might have lower impact on wage than raising it from 3 rd year undergraduate dropout to a BA degree graduate. Solutions-including non-linear terms such as logs, squares and interaction terms. by Kefyalew Endale (AAU) 7

8 Simple Linear regression CONT D Interpretations of β 1 in different models 1. Log-Log Model: both the dependent and independent variables are in logarithm lny = β 0 + β 1 lnx + ε d[lny] = β 1 d[lnx] Interpretation of β 1 : dy y = β 1 β 1 = dx x 100 dy y 100 dx x a. For β 1 >0: a 1% increase in X is associated with a β 1 % increase in Y b. For β 1 <0: a 1% increase in X is associated with a β 1 % decrease in Y by Kefyalew Endale (AAU) 8

9 Simple Linear Cont d 2. Log-Level Model: Y is in logarithm whereas X is in Level lny = β 0 + β 1 X + ε d[lny] = β 1 dx dy y = β 1dX Multiplying both sides by 100 and rearranging them yields Interpretation: 100*β 1 = 100 dy y dx =% Y dx for β 1 >0; a one 1 unit increase in X is associated with a β % increases Y For β 1 <0; a one unit increase in X is associated with a β 1 100% decrease in Y by Kefyalew Endale (AAU) 9

10 Simple Linear regression CONT D 3. Level-Log Model: Y is in level but X is in logarithm Y = β 0 + β 1 lnx + ε d[y] = β 1 d[lnx] dy = β 1 dx X β 1 = dy dx X Dividing both sides by 100 yields the following Interpretation: β 1 = dy 100 dx X 100 For β 1 >0; a 1% increase in X is associated with a β 1 unit increases in Y 100 For β 1 <0; a 1% increase in X is associated with a β 1 unit decrease in Y 100 by Kefyalew Endale (AAU) 10

11 Simple Linear regression Cont d Introduction to STATA STATA is a statistical package which is very helpful to estimate relationships between variables. Parts of STATA include Variable Window-it displays the list of variables in the dataset Command window-used to execute a single STATA command Results window (interface)-displays the outcomes of the STATA commands Review window-shows the list of executed commands Dofile editor-helps to save and execute several stata commands. Data editor-helps to inspect and edit the data Data browser-used to browse the data Variable manager-used to manage variables such as variable names and labels by Kefyalew Endale (AAU) 11

12 Simple Linear regression Cont d Dofile editor Data editor Variable manager Review window Data browser Result Window Variable window by Kefyalew Endale (AAU) 12

13 Simple Linear regression Basic stata commands des-to describe variables in the dataset sum-to summarize the variables sort-to sort the variables in increasing or decreasing order Eg. Sort hhid Label-to give value lables for the variables Eg. Label var hhid unique household identification number tab-to tabulate the variables Eg. tab toxen gen-to generate new variables from the existing variables Eg. gen hhagesq=hhage*hhage Reg-to regress one variable on another Eg. reg yield fertha by Kefyalew Endale (AAU) 13

14 Simple linear regression STATA sessions Using the wheat Yield data 1. describe the variables 2. Briefly interpret the summary of each variable (the number of observations, averages, minimum, and maximums) 3. Generate the natural logarithms of yield and fertilizer variables (note: if a variable has many zeros in its level, you should add one before transforming it into logarithm to reduce the numbers of missing values) 4. Estimate the following models for the regressions of yield (kg/ha) on fertilizer (kg/ha) a. The level-level model b. Log-log model c. Log-level model d. Level-log models 5. Interpret the coefficient of fertilizer (kg/ha) for each of the estimated model by Kefyalew Endale (AAU) 14

15 Simple Linear regression cont d VARIABLES (1) (2) (3) (4) level-level log-log log-level level-log fertha 3.937*** *** (6.720) (8.405) lnfertha 0.122*** 167.8*** (5.757) (4.144) Constant 1,070*** 6.626*** 6.795*** 903.0*** (10.06) (65.38) (123.0) (4.678) Observations R-squared t-statistics in parentheses, *** p<0.01, ** p<0.05, * p<0.1 by Kefyalew Endale (AAU) 15

16 Simple Linear regression CONT D Assumptions about ε [continued from equation 2.1] 1. E ε = 0: the average of the error term in the population is zero This is not a restrictive assumption so long as the model has an intercept term. 2. X and ε are uncorrelated or Cov Xε = 0 or their linear correlation is zero. This assumption, however, does not rule out non-linear correlation. For instance, Cov X 2 ε 0. Hence, the next assumption (which is a stronger one) is suggested. 3. The expected value of ε does not depend on the value of the regressor (or X). E ε/x = E(ε) The 3 rd assumption implies that ε is the same across all slices of the population E(ε/ 0 kg of fertilizer/ha) =E(ε/ non-zero kg of fertilizer/ha)= E(ε) by Kefyalew Endale (AAU) 16

17 Simple Linear regression CONT D If the conditional mean of ε is zero or E[ε/X]=0, then E[Y/X]= β 0 + β 1 X This equation says the average value of Y changes with X. It doesn t say Y=β 0 + β 1 X For any given value of X, the distribution of Y is centered about β 0 + β 1 X by Kefyalew Endale (AAU) 17

18 Simple Linear regression Cont d Eg. E[Yield kg/ha/fertilizer kg/ha]= Fertilizer kg/ha Then, as the quantity of fertilizer is 10 kg/ha, the expected value of yield becomes =1109kg/ha. This does not mean that the yield for every plot with a fertilizer quantity of 10kg/ha is 1109kg/ha The yield on some plots might be greater than 1109kg/ha and on some other plots it might be below 1109kg/ha. Whether the actual yield is above or below 1109kg/ha (at a fertilizer level of 10kg/ha) depends on the unobservable factors such as land quality. by Kefyalew Endale (AAU) 18

19 Simple Linear regression Cont d The Estimation Methods 1. OLS (Ordinary Least Squares)-is the most basic and most commonly used regression technique Y i = β 0 + β 1 X i + ε i We want to estimate Y i = β 0 + β 1 X i ^ denotes a sample estimate of the unobservable true population value OLS permits the estimation of β 0 and β 1 such that the sum of squared residuals (RSS) are minimized. e i = Y i Y i OLS minimizes i=1 n e 2 i or minimizes e e e2 n by Kefyalew Endale (AAU) 19

20 Simple Linear regression CONT D Why we use OLS? 1. It is easy to compute the parameters (one can easily compute them even by hand) 2. Theoretically appropriate: The method involves finding parameters which minimizes the sum square of errors. Because we want the difference between the actual and the predicted values of the dependent variables to be as small as possible. 3. Its useful properties a. The regression line passes through the averages of Y and X b. The sum of the residuals is zero c. OLS generated estimates can be said to be best under a set of restrictive assumptions. by Kefyalew Endale (AAU) 20

21 Simple Linear regression CONT D Min n i=1 e 2 i, wrt β 0, β 1 =Min n i=1 (Y i β 0 β 1 X i ) 2 Focs, RSS β 0 = -2 RSS n i=1 i=1 n (Y i β 0 β 1 X i ) =0.(1) = -2 Y β i β 0 β 1 X i X i =0.(2) 1 By solving these two FOCS simultaneously, we obtain the following: β 0 = Y β 1 X β 1 = i=1 n i=1 Y i X i n Y X n X i 2 n X 2 by Kefyalew Endale (AAU) 21

22 Simple Linear regression cont d Goodness of Fit; Evaluates how measures how well a regression model fits the data. The smaller the residual sum of squares, the better is the goodness of fit. In linear regressions, goodness of fit is measured by the coefficient of determination (R-square) Total sum of squares=estimated sum of squares + Residual sum of squares TSS=ESS+RSS TSS = n i=1 (Y i Y) 2 ESS= i=1 n ( Y i Y) 2 RSS= n i=1 (Y i Y i ) 2 R2= ESS = 1 RSS TSS TSS by Kefyalew Endale (AAU) 22

23 Simple Linear regression CONT D Statistical Significance: Estimated parameters are the estimates for the true relationships and there is uncertainty associated with the estimates. The uncertainty about each parameter is measured by its standard error or the standard deviation of the coefficient. The larger the standard errors, the larger the uncertainty in the parameter. Standard errors are used to test hypothesis. t-statistic which is the ratio of the coefficient to the standard error is used in the decision whether to reject the null hypothesis or not. Eg. The fertilizer regression Null hypothesis (H0): Fertilizer has no effect on wheat yield (β 1 = 0) Alternative hypothesis (H1): β 1 >0 Rule of thumb: Reject the null of the absolute value of the t-statistic is greater than 2 or when the coefficient is atleast twice as much as its standard deviation. by Kefyalew Endale (AAU) 23

24 Simple Linear regression Cont d In the wheat yield regression, the t-statistics for fertilizer is Which means the coefficient of fertilizer is 6.72 times the standard error. Hence, the null hypothesis is rejected. The results suggest that fertilizer use has significant positive impact on crop yield. by Kefyalew Endale (AAU) 24

ECON3150/4150 Spring 2015

ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2