# Nonlinear Regression Functions. SW Ch 8 1/54/

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Nonlinear Regression Functions SW Ch 8 1/54/

2 The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/

3 But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/

4 Nonlinear Regression General Ideas If a relation between Y and X is nonlinear: The effect on Y of a change in X depends on the value of X that is, the marginal effect of X is not constant A linear regression is mis-specified: the functional form is wrong The estimator of the effect on Y of X is biased: in general it isn t even right on average. The solution is to estimate a regression function that is nonlinear in X SW Ch 8 4/54/

5 The general nonlinear population regression function Y i = f(x 1i, X 2i,, X ki ) + u i, i = 1,, n Assumptions 1. E(u i X 1i, X 2i,, X ki ) = 0 (same) 2. (X 1i,, X ki, Y i ) are i.i.d. (same) 3. Big outliers are rare (same idea; the precise mathematical condition depends on the specific f) 4. No perfect multicollinearity (same idea; the precise statement depends on the specific f) SW Ch 8 5/54/

6 Outline 1. Nonlinear (polynomial) functions of one variable 2. Polynomial functions of multiple variables: Interactions 3. Application to the California Test Score data set 4. Addendum: Fun with logarithms SW Ch 8 6/54/

7 Nonlinear (Polynomial) Functions of a One RHS Variable Approximate the population regression function by a polynomial: 2 r Y i = X i + 2 X i + + r X i + u i This is just the linear multiple regression model except that the regressors are powers of X! Estimation, hypothesis testing, etc. proceeds as in the multiple regression model using OLS The coefficients are difficult to interpret, but the regression function itself is interpretable SW Ch 8 7/54/

8 Example: the TestScore Income relation Income i = average district income in the i th district (thousands of dollars per capita) Quadratic specification: TestScore i = Income i + 2 (Income i ) 2 + u i Cubic specification: TestScore i = Income i + 2 (Income i ) (Income i ) 3 + u i SW Ch 8 8/54/

9 Estimation of the quadratic specification in STATA generate avginc2 = avginc*avginc; Create a new regressor reg testscr avginc avginc2, r; Regression with robust standard errors Number of obs = 420 F( 2, 417) = Prob > F = R-squared = Root MSE = Robust testscr Coef. Std. Err. t P> t [95% Conf. Interval] avginc avginc _cons Test the null hypothesis of linearity against the alternative that the regression function is a quadratic. SW Ch 8 9/54/

10 Interpreting the estimated regression function: (a) Plot the predicted values TestScore = Income i (Income i ) 2 (2.9) (0.27) (0.0048) SW Ch 8 10/54/

11 Interpreting the estimated regression function, ctd: (b) Compute effects for different values of X TestScore = Income i (Income i ) 2 (2.9) (0.27) (0.0048) Predicted change in TestScore for a change in income from \$5,000 per capita to \$6,000 per capita: TestScore = ( ) = 3.4 SW Ch 8 11/54/

12 TestScore = Income i (Income i ) 2 Predicted effects for different values of X: Change in Income (\$1000 per capita) TestScore from 5 to from 25 to from 45 to The effect of a change in income is greater at low than high income levels (perhaps, a declining marginal benefit of an increase in school budgets?) Caution! What is the effect of a change from 65 to 66? Don t extrapolate outside the range of the data! SW Ch 8 12/54/

13 Estimation of a cubic specification in STATA gen avginc3 = avginc*avginc2; Create the cubic regressor reg testscr avginc avginc2 avginc3, r; Regression with robust standard errors Number of obs = 420 F( 3, 416) = Prob > F = R-squared = Root MSE = Robust testscr Coef. Std. Err. t P> t [95% Conf. Interval] avginc avginc avginc e _cons SW Ch 8 13/54/

14 Testing the null hypothesis of linearity, against the alternative that the population regression is quadratic and/or cubic, that is, it is a polynomial of degree up to 3: H 0 : population coefficients on Income 2 and Income 3 = 0 H 1 : at least one of these coefficients is nonzero. test avginc2 avginc3; Execute the test command after running the regression ( 1) avginc2 = 0.0 ( 2) avginc3 = 0.0 F( 2, 416) = Prob > F = The hypothesis that the population regression is linear is rejected at the 1% significance level against the alternative that it is a polynomial of degree up to 3. SW Ch 8 14/54/

15 Summary: polynomial regression functions 2 r Y i = X i + 2 X i + + r X i + u i Estimation: by OLS after defining new regressors Coefficients have complicated interpretations To interpret the estimated regression function: o plot predicted values as a function of x o compute predicted Y/ X at different values of x Hypotheses concerning degree r can be tested by t- and F- tests on the appropriate (blocks of) variable(s). Choice of degree r o plot the data; t- and F-tests, check sensitivity of estimated effects; judgment. o Or use model selection criteria (later) SW Ch 8 15/54/

16 Polynomials in Multiple Variables: Interactions Perhaps a class size reduction is more effective in some circumstances than in others Perhaps smaller classes help more if there are many English learners, who need individual attention TestScore That is, might depend on PctEL STR Y More generally, might depend on X 2 X 1 How to model such interactions between X 1 and X 2? We first consider binary X s, then continuous X s SW Ch 8 16/54/

17 (a) Interactions between two binary variables Y i = D 1i + 2 D 2i + u i D 1i, D 2i are binary 1 is the effect of changing D 1 =0 to D 1 =1. In this specification, this effect doesn t depend on the value of D 2. To allow the effect of changing D 1 to depend on D 2, include the interaction term D 1i D 2i as a regressor: Y i = D 1i + 2 D 2i + 3 (D 1i D 2i ) + u i SW Ch 8 17/54/

18 Interpreting the coefficients Y i = D 1i + 2 D 2i + 3 (D 1i D 2i ) + u i The effect of D 1 depends on d 2 (what we wanted) 3 = increment to the effect of D 1, when D 2 = 1 SW Ch 8 18/54/

19 Example: TestScore, STR, English learners Let 1 if STR 20 1 if PctEL l0 HiSTR = and HiEL = 0 if STR 20 0 if PctEL 10 TestScore = HiEL 1.9HiSTR 3.5(HiSTR HiEL) (1.4) (2.3) (1.9) (3.1) Effect of HiSTR when HiEL = 0 is 1.9 Effect of HiSTR when HiEL = 1 is = 5.4 Class size reduction is estimated to have a bigger effect when the percent of English learners is large This interaction isn t statistically significant: t = 3.5/3.1 SW Ch 8 19/54/

20 (b) Interactions between continuous and binary variables Y i = D i + 2 X i + u i D i is binary, X is continuous As specified above, the effect on Y of X (holding constant D) = 2, which does not depend on D To allow the effect of X to depend on D, include the interaction term D i X i as a regressor: Y i = D i + 2 X i + 3 (D i X i ) + u i SW Ch 8 20/54/

21 Binary-continuous interactions: the two regression lines Y i = D i + 2 X i + 3 (D i X i ) + u i Observations with D i = 0 (the D = 0 group): Y i = X i + u i The D=0 regression line Observations with D i = 1 (the D = 1 group): Y i = X i + 3 X i + u i = ( ) + ( )X i + u i The D=1 regression line SW Ch 8 21/54/

22 Binary-continuous interactions, ctd. SW Ch 8 22/54/

23 Interpreting the coefficients Y i = D i + 2 X i + 3 (D i X i ) + u i 1 = increment to intercept when D=1 3 = increment to slope when D = 1 SW Ch 8 23/54/

24 Example: TestScore, STR, HiEL (=1 if PctEL 10) TestScore = STR + 5.6HiEL 1.28(STR HiEL) (11.9) (0.59) (19.5) (0.97) When HiEL = 0: TestScore = STR When HiEL = 1, TestScore = STR STR = STR Two regression lines: one for each HiSTR group. Class size reduction is estimated to have a larger effect when the percent of English learners is large. SW Ch 8 24/54/

25 Example, ctd: Testing hypotheses TestScore = STR + 5.6HiEL 1.28(STR HiEL) (11.9) (0.59) (19.5) (0.97) The two regression lines have the same slope the coefficient on STR HiEL is zero: t = 1.28/0.97 = 1.32 The two regression lines have the same intercept the coefficient on HiEL is zero: t = 5.6/19.5 = 0.29 The two regression lines are the same population coefficient on HiEL = 0 and population coefficient on STR HiEL = 0: F = (p-value <.001)!! We reject the joint hypothesis but neither individual hypothesis (how can this be?) SW Ch 8 25/54/

26 (c) Interactions between two continuous variables Y i = X 1i + 2 X 2i + u i X 1, X 2 are continuous As specified, the effect of X 1 doesn t depend on X 2 As specified, the effect of X 2 doesn t depend on X 1 To allow the effect of X 1 to depend on X 2, include the interaction term X 1i X 2i as a regressor: Y i = X 1i + 2 X 2i + 3 (X 1i X 2i ) + u i SW Ch 8 26/54/

27 Interpreting the coefficients: Y i = X 1i + 2 X 2i + 3 (X 1i X 2i ) + u i The effect of X 1 depends on X 2 (what we wanted) 3 = increment to the effect of X 1 from a unit change in X 2 SW Ch 8 27/54/

28 Example: TestScore, STR, PctEL TestScore = STR 0.67PctEL (STR PctEL), (11.8) (0.59) (0.37) (0.019) The estimated effect of class size reduction is nonlinear because the size of the effect itself depends on PctEL: TestScore = PctEL STR PctEL TestScore STR % = 1.10 SW Ch 8 28/54/

29 Example, ctd: hypothesis tests TestScore = STR 0.67PctEL (STR PctEL), (11.8) (0.59) (0.37) (0.019) Does population coefficient on STR PctEL = 0? t =.0012/.019 =.06 can t reject null at 5% level Does population coefficient on STR = 0? t = 1.12/0.59 = 1.90 can t reject null at 5% level Do the coefficients on both STR and STR PctEL = 0? F = 3.89 (p-value =.021) reject null at 5% level(!!) (Why?) SW Ch 8 29/54/

30 Application: Nonlinear Effects on Test Scores of the Student-Teacher Ratio Nonlinear specifications let us examine more nuanced questions about the Test score STR relation, such as: 1. Are there nonlinear effects of class size reduction on test scores? (Does a reduction from 35 to 30 have same effect as a reduction from 20 to 15?) 2. Are there nonlinear interactions between PctEL and STR? (Are small classes more effective when there are many English learners?) SW Ch 8 30/54/

31 Strategy for Question #1 (different effects for different STR?) Estimate linear and nonlinear functions of STR, holding constant relevant demographic variables o PctEL o Income (remember the nonlinear TestScore-Income relation!) o LunchPCT (fraction on free/subsidized lunch) See whether adding the nonlinear terms makes an economically important quantitative difference ( economic or real-world importance is different than statistically significant) Test for whether the nonlinear terms are significant SW Ch 8 31/54/

32 Strategy for Question #2 (interactions between PctEL and STR?) Estimate linear and nonlinear functions of STR, interacted with PctEL. If the specification is nonlinear (with STR, STR 2, STR 3 ), then you need to add interactions with all the terms so that the entire functional form can be different, depending on the level of PctEL. We will use a binary-continuous interaction specification by adding HiEL STR, HiEL STR 2, and HiEL STR 3. SW Ch 8 32/54/

33 What is a good base specification? The TestScore Income relation: The logarithmic specification is better behaved near the extremes of the sample, especially for large values of income. SW Ch 8 33/54/

34 SW Ch 8 34/54/

35 Tests of joint hypotheses: What can you conclude about question #1? About question #2? SW Ch 8 35/54/

36 Interpreting the regression functions via plots: First, compare the linear and nonlinear specifications: SW Ch 8 36/54/

37 Next, compare the regressions with interactions: SW Ch 8 37/54/

38 Addendum Fun with logarithms Y and/or X is transformed by taking its logarithm this gives a percentages interpretation that makes sense in many applications SW Ch 8 38/54/

39 2. Logarithmic functions of Y and/or X ln(x) = the natural logarithm of X Logarithmic transforms permit modeling relations in percentage terms (like elasticities), rather than linearly. Key result (recall from calculus): For small changes, the change in the log is approximately the percent change (expressed as a decimal). SW Ch 8 39/54/

40 The three log regression specifications: Case Population regression function I. linear-log Y i = ln(x i ) + u i II. log-linear ln(y i ) = X i + u i III. log-log ln(y i ) = ln(x i ) + u i The interpretation of the slope coefficient differs in each case. The interpretation is found by applying the general before and after rule: figure out the change in Y for a given change in X. Each case has a natural interpretation (for small changes in X) SW Ch 8 40/54/

41 I. Linear-log population regression function a 1% increase in X (multiplying X by 1.01) is associated with a.01 1 change in Y. (1% increase in X.01 increase in ln(x).01 1 increase in Y) SW Ch 8 41/54/

42 Example: TestScore vs. ln(income) First define the new regressor, ln(income) The model is now linear in ln(income), so the linear-log model can be estimated by OLS: TestScore = ln(income i ) (3.8) (1.40) so a 1% increase in Income is associated with an increase in TestScore of 0.36 points on the test. SW Ch 8 42/54/

43 The linear-log and cubic regression functions SW Ch 8 43/54/

44 II. Log-linear population regression function a change in X by one unit ( X = 1) is associated with a % change in Y 1 unit increase in X 1 increase in ln(y) % increase in Y SW Ch 8 44/54/

45 III. Log-log population regression function a 1% change in X is associated with a 1 % change in Y. In the log-log specification, 1 has the interpretation of an elasticity. SW Ch 8 45/54/

46 Example: ln(testscore) vs. ln( Income) First define a new dependent variable, ln(testscore), and the new regressor, ln(income) The model is now a linear regression of ln(testscore) against ln(income), which can be estimated by OLS: ln( TestScore ) = ln(income i ) (0.006) (0.0021) An 1% increase in Income is associated with an increase of.0554% in TestScore (Income up by a factor of 1.01, TestScore up by a factor of ) SW Ch 8 46/54/

47 Example: ln( TestScore) vs. ln( Income), ctd. ln( TestScore ) = ln(income i ) (0.006) (0.0021) For example, suppose income increases from \$10,000 to \$11,000, or by 10%. Then TestScore increases by approximately % =.554%. If TestScore = 650, this corresponds to an increase of = 3.6 points. How does this compare to the log-linear model? SW Ch 8 47/54/

48 The log-linear and log-log specifications: Note vertical axis Neither seems to fit as well as the cubic or linear-log, at least based on visual inspection (formal comparison is difficult because the dependent variables differ) SW Ch 8 48/54/

49 Summary: Logarithmic transformations Three cases, differing in whether Y and/or X is transformed by taking logarithms. The regression is linear in the new variable(s) ln(y) and/or ln(x), and the coefficients can be estimated by OLS. Hypothesis tests and confidence intervals are now implemented and interpreted as usual. The interpretation of 1 differs from case to case. The choice of specification (functional form) should be guided by judgment (which interpretation makes the most sense in your application?), tests, and plotting predicted values SW Ch 8 49/54/

50 Other nonlinear functions (and nonlinear least squares) The foregoing regression functions have limitations Polynomial: test score can decrease with income Linear-log: test score increases with income, but without bound Here is a nonlinear function in which Y always increases with X and there is a maximum (asymptote) value of Y: 1 Y = 0 X e 0, 1, and are unknown parameters. This is called a negative exponential growth curve. The asymptote as X is 0. SW Ch 8 50/54/

51 Negative exponential growth We want to estimate the parameters of 1X Y i = e i u i 0 Compare to linear-log or cubic models: Y i = ln(x i ) + u i 2 3 Y i = X i + 2 X i + 2 X i + u i Linear-log and polynomial models are linear in the parameters 0 and 1, but the negative exponential model is not. SW Ch 8 51/54/

52 Nonlinear Least Squares Models that are linear in the parameters can be estimated by OLS. Models that are nonlinear in one or more parameters can be estimated by nonlinear least squares (NLS) (but not by OLS) What is the NLS problem for the proposed specification? This is a nonlinear minimization problem (a hill-climbing problem). How could you solve this? o Guess and check o There are better ways o Implementation... SW Ch 8 52/54/

53 Negative exponential growth; RMSE = Linear-log; RMSE = (oh well ) SW Ch 8 53/54/

54 SW Ch 8 54/54/

### Linear Regression with One Regressor

Linear Regression with One Regressor Michael Ash Lecture 10 Analogy to the Mean True parameter µ Y β 0 and β 1 Meaning Central tendency Intercept and slope E(Y ) E(Y X ) = β 0 + β 1 X Data Y i (X i, Y

### IAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results

IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the

### REGRESSION LINES IN STATA

REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression

### 2. Linear regression with multiple regressors

2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions

### Discussion Section 4 ECON 139/239 2010 Summer Term II

Discussion Section 4 ECON 139/239 2010 Summer Term II 1. Let s use the CollegeDistance.csv data again. (a) An education advocacy group argues that, on average, a person s educational attainment would increase

### Regression Analysis. Data Calculations Output

Regression Analysis In an attempt to find answers to questions such as those posed above, empirical labour economists use a useful tool called regression analysis. Regression analysis is essentially a

### August 2012 EXAMINATIONS Solution Part I

August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

### Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software

STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used

### MULTIPLE REGRESSION EXAMPLE

MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if

### Econ 371 Problem Set #3 Answer Sheet

Econ 371 Problem Set #3 Answer Sheet 4.3 In this question, you are told that a OLS regression analysis of average weekly earnings yields the following estimated model. AW E = 696.7 + 9.6 Age, R 2 = 0.023,

### Econ 371 Problem Set #3 Answer Sheet

Econ 371 Problem Set #3 Answer Sheet 4.1 In this question, you are told that a OLS regression analysis of third grade test scores as a function of class size yields the following estimated model. T estscore

### Semi-Log Model. Semi-Log Model

Semi-Log Model The slope coefficient measures the relative change in Y for a given absolute change in the value of the explanatory variable (t). Semi-Log Model Using calculus: ln Y b2 = t 1 Y = Y t Y =

### Nonlinear relationships Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015

Nonlinear relationships Richard Williams, University of Notre Dame, http://www.nd.edu/~rwilliam/ Last revised February, 5 Sources: Berry & Feldman s Multiple Regression in Practice 985; Pindyck and Rubinfeld

### Linear Regression Models with Logarithmic Transformations

Linear Regression Models with Logarithmic Transformations Kenneth Benoit Methodology Institute London School of Economics kbenoit@lse.ac.uk March 17, 2011 1 Logarithmic transformations of variables Considering

### In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a

Math 143 Inference on Regression 1 Review of Linear Regression In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a bivariate data set (i.e., a list of cases/subjects

### ECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2

University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages

### Correlation and Regression

Correlation and Regression Scatterplots Correlation Explanatory and response variables Simple linear regression General Principles of Data Analysis First plot the data, then add numerical summaries Look

### where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.

Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes

### ECON Introductory Econometrics. Lecture 17: Experiments

ECON4150 - Introductory Econometrics Lecture 17: Experiments Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 13 Lecture outline 2 Why study experiments? The potential outcome framework.

### Rockefeller College University at Albany

Rockefeller College University at Albany PAD 705 Handout: Hypothesis Testing on Multiple Parameters In many cases we may wish to know whether two or more variables are jointly significant in a regression.

### The data is more curved than linear, but not so much that linear regression is not an option. The line of best fit for linear regression is

CHAPTER 5 Nonlinear Models 1. Quadratic Function Models Consider the following table of the number of Americans that are over 100 in various years. Year Number (in 1000 s) 1994 50 1996 56 1998 65 2000

### Lecture 15. Endogeneity & Instrumental Variable Estimation

Lecture 15. Endogeneity & Instrumental Variable Estimation Saw that measurement error (on right hand side) means that OLS will be biased (biased toward zero) Potential solution to endogeneity instrumental

### Quantitative Methods for Economics Tutorial 9. Katherine Eyal

Quantitative Methods for Economics Tutorial 9 Katherine Eyal TUTORIAL 9 4 October 2010 ECO3021S Part A: Problems 1. In Problem 2 of Tutorial 7, we estimated the equation ŝleep = 3, 638.25 0.148 totwrk

### Quick Stata Guide by Liz Foster

by Liz Foster Table of Contents Part 1: 1 describe 1 generate 1 regress 3 scatter 4 sort 5 summarize 5 table 6 tabulate 8 test 10 ttest 11 Part 2: Prefixes and Notes 14 by var: 14 capture 14 use of the

### Department of Economics, Session 2012/2013. EC352 Econometric Methods. Exercises from Week 03

Department of Economics, Session 01/013 University of Essex, Autumn Term Dr Gordon Kemp EC35 Econometric Methods Exercises from Week 03 1 Problem P3.11 The following equation describes the median housing

### Linearizing Data. Lesson3. United States Population

Lesson3 Linearizing Data You may have heard that the population of the United States is increasing exponentially. The table and plot below give the population of the United States in the census years 19

### Econ 371 Problem Set #4 Answer Sheet. P rice = (0.485)BDR + (23.4)Bath + (0.156)Hsize + (0.002)LSize + (0.090)Age (48.

Econ 371 Problem Set #4 Answer Sheet 6.5 This question focuses on what s called a hedonic regression model; i.e., where the sales price of the home is regressed on the various attributes of the home. The

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### Statistical Modelling in Stata 5: Linear Models

Statistical Modelling in Stata 5: Linear Models Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 08/11/2016 Structure This Week What is a linear model? How

### Basic Econometrics Tools Correlation and Regression Analysis

Basic Econometrics Tools Correlation and Regression Analysis Christopher Grigoriou Executive MBA HEC Lausanne 2007/2008 1 A collector of antique grandfather clocks wants to know if the price received for

### Lectures 8, 9 & 10. Multiple Regression Analysis

Lectures 8, 9 & 0. Multiple Regression Analysis In which you learn how to apply the principles and tests outlined in earlier lectures to more realistic models involving more than explanatory variable and

### The Simple Linear Regression Model: Specification and Estimation

Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### Inference for Regression

Simple Linear Regression Inference for Regression The simple linear regression model Estimating regression parameters; Confidence intervals and significance tests for regression parameters Inference about

### Department of Economics Session 2012/2013. EC352 Econometric Methods. Solutions to Exercises from Week 10 + 0.0077 (0.052)

Department of Economics Session 2012/2013 University of Essex Spring Term Dr Gordon Kemp EC352 Econometric Methods Solutions to Exercises from Week 10 1 Problem 13.7 This exercise refers back to Equation

### Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### How Do We Test Multiple Regression Coefficients?

How Do We Test Multiple Regression Coefficients? Suppose you have constructed a multiple linear regression model and you have a specific hypothesis to test which involves more than one regression coefficient.

### Introduction to Stata

Introduction to Stata September 23, 2014 Stata is one of a few statistical analysis programs that social scientists use. Stata is in the mid-range of how easy it is to use. Other options include SPSS,

### Multiple Regression - Selecting the Best Equation An Example Techniques for Selecting the "Best" Regression Equation

Multiple Regression - Selecting the Best Equation When fitting a multiple linear regression model, a researcher will likely include independent variables that are not important in predicting the dependent

### Multinomial and Ordinal Logistic Regression

Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit August 22, 2012 Regression with categorical dependent variables When the dependent variable is categorical,

### Calculus Card Matching

Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

### MODEL I: DRINK REGRESSED ON GPA & MALE, WITHOUT CENTERING

Interpreting Interaction Effects; Interaction Effects and Centering Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015 Models with interaction effects

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### Statistics 104 Final Project A Culture of Debt: A Study of Credit Card Spending in America TF: Kevin Rader Anonymous Students: LD, MH, IW, MY

Statistics 104 Final Project A Culture of Debt: A Study of Credit Card Spending in America TF: Kevin Rader Anonymous Students: LD, MH, IW, MY ABSTRACT: This project attempted to determine the relationship

### Handling missing data in Stata a whirlwind tour

Handling missing data in Stata a whirlwind tour 2012 Italian Stata Users Group Meeting Jonathan Bartlett www.missingdata.org.uk 20th September 2012 1/55 Outline The problem of missing data and a principled

### Questions and Answers on Hypothesis Testing and Confidence Intervals

Questions and Answers on Hypothesis Testing and Confidence Intervals L. Magee Fall, 2008 1. Using 25 observations and 5 regressors, including the constant term, a researcher estimates a linear regression

### Stata Walkthrough 4: Regression, Prediction, and Forecasting

Stata Walkthrough 4: Regression, Prediction, and Forecasting Over drinks the other evening, my neighbor told me about his 25-year-old nephew, who is dating a 35-year-old woman. God, I can t see them getting

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

### Logs Transformation in a Regression Equation

Fall, 2001 1 Logs as the Predictor Logs Transformation in a Regression Equation The interpretation of the slope and intercept in a regression change when the predictor (X) is put on a log scale. In this

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### Multicollinearity Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015

Multicollinearity Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015 Stata Example (See appendices for full example).. use http://www.nd.edu/~rwilliam/stats2/statafiles/multicoll.dta,

### Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA

Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA Introduction A common question faced by many actuaries when selecting loss development factors is whether to base the selected

### Interaction effects and group comparisons Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015

Interaction effects and group comparisons Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015 Note: This handout assumes you understand factor variables,

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

### AP * Statistics Review. Linear Regression

AP * Statistics Review Linear Regression Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production

### Marginal Effects for Continuous Variables Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 21, 2015

Marginal Effects for Continuous Variables Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 21, 2015 References: Long 1997, Long and Freese 2003 & 2006 & 2014,

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### Using R for Linear Regression

Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### Problems with OLS Considering :

Problems with OLS Considering : we assume Y i X i u i E u i 0 E u i or var u i E u i u j 0orcov u i,u j 0 We have seen that we have to make very specific assumptions about u i in order to get OLS estimates

### Lecture 13. Use and Interpretation of Dummy Variables. Stop worrying for 1 lecture and learn to appreciate the uses that dummy variables can be put to

Lecture 13. Use and Interpretation of Dummy Variables Stop worrying for 1 lecture and learn to appreciate the uses that dummy variables can be put to Using dummy variables to measure average differences

### Extending Hypothesis Testing. p-values & confidence intervals

Extending Hypothesis Testing p-values & confidence intervals So far: how to state a question in the form of two hypotheses (null and alternative), how to assess the data, how to answer the question by

### Simple Linear Regression

STAT 101 Dr. Kari Lock Morgan Simple Linear Regression SECTIONS 9.3 Confidence and prediction intervals (9.3) Conditions for inference (9.1) Want More Stats??? If you have enjoyed learning how to analyze

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Multiple Linear Regression

Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is

### Standard errors of marginal effects in the heteroskedastic probit model

Standard errors of marginal effects in the heteroskedastic probit model Thomas Cornelißen Discussion Paper No. 320 August 2005 ISSN: 0949 9962 Abstract In non-linear regression models, such as the heteroskedastic

### Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression

Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009

HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Introduction 2. A General Formulation 3. Truncated Normal Hurdle Model 4. Lognormal

### Midterm 1. Solutions

Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### SELF-TEST: SIMPLE REGRESSION

ECO 22000 McRAE SELF-TEST: SIMPLE REGRESSION Note: Those questions indicated with an (N) are unlikely to appear in this form on an in-class examination, but you should be able to describe the procedures

### Economics 345 Applied Econometrics

Economics 345 Applied Econometrics Lab 3: Some Useful Functional Forms for Regression Analysis Prof: Martin Farnham TAs: Joffré Leroux, Rebecca Wortzman, Yiying Yang Open EViews, and open the EViews workfile,

### A Primer on Forecasting Business Performance

A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Failure to take the sampling scheme into account can lead to inaccurate point estimates and/or flawed estimates of the standard errors.

Analyzing Complex Survey Data: Some key issues to be aware of Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 24, 2015 Rather than repeat material that is

### Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

### Lecture 16: Logistic regression diagnostics, splines and interactions. Sandy Eckel 19 May 2007

Lecture 16: Logistic regression diagnostics, splines and interactions Sandy Eckel seckel@jhsph.edu 19 May 2007 1 Logistic Regression Diagnostics Graphs to check assumptions Recall: Graphing was used to

### Introduction to Time Series Regression and Forecasting

Introduction to Time Series Regression and Forecasting (SW Chapter 14) Time series data are data collected on the same observational unit at multiple time periods Aggregate consumption and GDP for a country

### Using Stata 11 & higher for Logistic Regression Richard Williams, University of Notre Dame, Last revised March 28, 2015

Using Stata 11 & higher for Logistic Regression Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised March 28, 2015 NOTE: The routines spost13, lrdrop1, and extremes are

### Generalized Linear Models

Generalized Linear Models We have previously worked with regression models where the response variable is quantitative and normally distributed. Now we turn our attention to two types of models where the

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### 11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Interaction effects between continuous variables (Optional)

Interaction effects between continuous variables (Optional) Richard Williams, University of Notre Dame, http://www.nd.edu/~rwilliam/ Last revised February 0, 05 This is a very brief overview of this somewhat

### Cointegration and the ECM

Cointegration and the ECM Two nonstationary time series are cointegrated if they tend to move together through time. For instance, we have established that the levels of the Fed Funds rate and the 3-year

### DETERMINANTS OF CAPITAL ADEQUACY RATIO IN SELECTED BOSNIAN BANKS

DETERMINANTS OF CAPITAL ADEQUACY RATIO IN SELECTED BOSNIAN BANKS Nađa DRECA International University of Sarajevo nadja.dreca@students.ius.edu.ba Abstract The analysis of a data set of observation for 10

### Nominal and Real U.S. GDP 1960-2001

Problem Set #5-Key Sonoma State University Dr. Cuellar Economics 318- Managerial Economics Use the data set for gross domestic product (gdp.xls) to answer the following questions. (1) Show graphically

### ESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics

ESTIMATING AVERAGE TREATMENT EFFECTS: IV AND CONTROL FUNCTIONS, II Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Quantile Treatment Effects 2. Control Functions

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### LOGIT AND PROBIT ANALYSIS

LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y

### Larson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN

ALG B Algebra II, Second Semester #PR-0, BK-04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO

### Simultaneous Equation Models As discussed last week, one important form of endogeneity is simultaneity. This arises when one or more of the

Simultaneous Equation Models As discussed last week, one important form of endogeneity is simultaneity. This arises when one or more of the explanatory variables is jointly determined with the dependent