# Wooldridge, Introductory Econometrics, 4th ed. Multiple regression analysis:

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Wooldridge, Introductory Econometrics, 4th ed. Chapter 4: Inference Multiple regression analysis: We have discussed the conditions under which OLS estimators are unbiased, and derived the variances of these estimators under the Gauss- Markov assumptions. The Gauss-Markov theorem establishes that OLS estimators have the smallest variance of any linear unbiased estimators of the population parameters. We must now more fully characterize the sampling distribution of the OLS estimators beyond its mean and variance so that we may test hypotheses on the population parameters. To make the sampling distribution tractable, we add an assumption on the distribution of the errors:

2 Proposition 1 MLR6 (Normality) The population error u is independent of the explanatory variables x 1,.., x k and is normally distributed with zero mean and constant variance: u N ( 0, σ 2). This is a much stronger assumption than we have previously made on the distribution of the errors. The assumption of normality, as we have stated it, subsumes both the assumption of the error process being independent of the explanatory variables, and that of homoskedasticity. For cross-sectional regression analysis, these six assumptions define the classical linear model. The rationale for normally distributed errors is often phrased in terms of the many factors influencing y being additive, appealing to the Central Limit Theorem to suggest that the sum of a large number of random factors will be normally distributed. Although we might have reason in a particular context

3 to doubt this rationale, we usually use it as a working hypothesis. Various transformations such as taking the logarithm of the dependent variable are often motivated in terms of their inducing normality in the resulting errors. What is the importance of assuming normality for the error process? Under the assumptions of the classical linear model, normally distributed errors give rise to normally distributed OLS estimators: b j N ( β j, V ar ( )) b j (1) which will then imply that: ( bj β j ) σ bj N (0, 1) (2) This follows since each of the b j can be written as a linear combination of the errors in the

4 sample. Since we assume that the errors are independent, identically distributed normal random variates, any linear combination of those errors is also normally distributed. We may also show that any linear combination of the b j is also normally distributed, and a subset of these estimators has a joint normal distribution. These properties will come in handy in formulating tests on the coefficient vector. We may also show that the OLS estimators will be approximately normally distributed (at least in large samples), even if the underlying errors are not normally distributed. Testing an hypothesis on a single β j To test hypotheses about a single population parameter, we start with the model containing k regressors: y = β 0 + β 1 x 1 + β 2 x β k x k + u (3)

5 Under the classical linear model assumptions, a test statistic formed from the OLS estimates may be expressed as: ( bj β j ) s bj t n k 1 (4) Why does this test statistic differ from (2) above? In that expression, we considered the variance of b j as an expression including σ, the unknown standard deviation of the error term (that is, σ 2 ). In this operational test statistic (4), we have replaced σ with a consistent estimate, s. That additional source of sampling variation requires the switch from the standard normal distribution to the t distribution, with (n k 1) degrees of freedom. Where n is not all that large relative to k, the resulting t distribution will have considerably fatter tails than the standard normal. Where (n k 1) is a

6 large number greater than 100, for instance the t distribution will essentially be the standard normal. The net effect is to make the critical values larger for a finite sample, and raise the threshold at which we will conclude that there is adequate evidence to reject a particular hypothesis. The test statistic (4) allows us to test hypotheses regarding the population parameter β j : in particular, to test the null hypothesis H 0 : β j = 0 (5) for any of the regression parameters. The t-statistic used for this test is merely that printed on the output when you run a regression in Stata or any other program: the ratio of the estimated coefficient to its estimated standard error. If the null hypothesis is to be rejected, the t-stat must be larger (in absolute value) than the critical point on the t- distribution. The t-stat will have the same

7 sign as the estimated coefficient, since the standard error is always positive. Even if β j is actually zero in the population, a sample estimate of this parameter b j will never equal exactly zero. But when should we conclude that it could be zero? When its value cannot be distinguished from zero. There will be cause to reject this null hypothesis if the value, scaled by its standard error, exceeds the threshold. For a two-tailed test, there will be reason to reject the null if the t-stat takes on a large negative value or a large positive value; thus we reject in favor of the alternative hypothesis (of β j 0) in either case. This is a two-sided alternative, giving rise to a two-tailed test. If the hypothesis is to be tested at, e.g., the 95% level of confidence, we use critical values from the t-distribution which isolate 2.5% in each tail, for a total of 5% of the mass of the distribution. When using a computer program to calculate regression estimates, we usually are

8 given the p-value of the estimate that is, the tail probability corresponding to the coefficient s t-value. The p-value may usefully be considered as the probability of observing a t- statistic as extreme as that shown if the null hypothesis is true. If the t-value was equal to, e.g., the 95% critical value, the p-value would be exactly If the t-value was higher, the p-value would be closer to zero, and vice versa. Thus, we are looking for small p-values as indicative of rejection. A p-value of 0.92, for instance, corresponds to an hypothesis that can be rejected at the 8% level of confidence thus quite irrelevant, since we would expect to find a value that large 92% of the time under the null hypothesis. On the other hand, a p-value of 0.08 will reject at the 90% level, but not at the 95% level; only 8% of the time would we expect to find a t-statistic of that magnitude if H 0 was true.

9 What if we have a one-sided alternative? For instance, we may phrase the hypothesis of interest as: H 0 : β j > 0 (6) H A : β j 0 Here, we must use the appropriate critical point on the t-distribution to perform this test at the same level of confidence. If the point estimate b j is positive, then we do not have cause to reject the null. If it is negative, we may have cause to reject the null if it is a sufficiently large negative value. The critical point should be that which isolates 5% of the mass of the distribution in that tail (for a 95% level of confidence). This critical value will be smaller (in absolute value) than that corresponding to a two-tailed test, which isolates only 2.5% of the mass in that tail. The computer program always provides you with a p-value for a twotailed test; if the p-value is 0.08, for instance,

10 it corresponds to a one-tailed p-value of 0.04 (that being the mass in that tail). Testing other hypotheses about β j Every regression output includes the information needed to test the two-tailed or one-tailed hypotheses that a population parameter equals zero. What if we want to test a different hypothesis about the value of that parameter? For instance, we would not consider it sensible for the mpc for a consumer to be zero, but we might have an hypothesized value (of, say, 0.8) implied by a particular theory of consumption. How might we test this hypothesis? If the null is stated as: H 0 : β j = a j (7) where a j is the hypothesized value, then the appropriate test statistic becomes:

11 ( bj a j ) s bj t n k 1 (8) and we may simply calculate that quantity and compare it to the appropriate point on the t- distribution. Most computer programs provide you with assistance in this effort; for instance, if we believed that a j, the coefficient on bdrms, should be equal to \$20,000 in a regression of house prices on square footage and bdrms (e.g. using HPRICE1), we would use Stata s test command: regress price bdrms sqrft test bdrms=20000 where we use the name of the variable as a shorthand for the name of the coefficient on that variable. Stata, in that instance, presents us with:

12 ( 1) bdrms = F( 1, 85) = 0.26 Prob > F = making use of an F-statistic, rather than a t- statistic, to perform this test. In this particular case of an hypothesis involving a single regression coefficient we may show that this F-statistic is merely the square of the associated t-statistic. The p-value would be the same in either case. The estimated coefficient is , with an estimated standard error of Plugging in these values to (8) yields a t-statistic:. di (_b[bdrms]-20000)/_se[bdrms] which, squared, is the F-statistic shown by the test command. Just as with tests against a

13 null hypothesis of zero, the results of the test command may be used for one-tailed tests as well as two-tailed tests; then, the magnitude of the coefficient matters (i.e. the fact that the estimated coefficient is about \$15,000 means we would never reject a null that it is less than \$20,000), and the p-value must be adjusted for one tail. Any number of test commands may be given after a regress command in Stata, testing different hypotheses about the coefficients. Confidence intervals As we discussed in going over Appendix C, we may use the point estimate and its estimated standard error to calculate an hypothesis test on the underlying population parameter, or we may form a confidence interval for that parameter. Stata makes that easy in a regression

14 context by providing the 95% confidence interval for every estimated coefficient. If you want to use some other level of significance, you may either use the level() option on regress (e.g. regress price bdrms sqrft, level(90)) or you may change the default level for this run with set level. All further regressions will report confidence intervals with that level of confidence. To connect this concept to that of the hypothesis test, consider that in the above example the 95% confidence interval for β bdrms extended from to ; thus, an hypothesis test with the null that β bdrms takes on any value in this interval (including zero) will not lead to a rejection. Testing hypotheses about a single linear combination of the parameters Economic theory will often suggest that a particular linear combination of parameters should

15 take on a certain value: for instance, in a Cobb-Douglas production function, that the slope coefficients should sum to one in the case of constant returns to scale (CRT S): Q = AL β 1K β 2E β 3 (9) log Q = log A + β 1 log L + β 2 log K + β 3 log E + υ where K, L, E are the factors capital, labor, and energy, respectively. We have added an error term to the double-log-transformed version of this model to represent it as an empirical relationship. The hypothesis of CRT S may be stated as: H 0 : β 1 + β 2 + β 3 = 1 (10) The test statistic for this hypothesis is quite straightforward: (b 1 + b 2 + b 3 1) s b1 +b 2 +b 3 t n k 1 (11)

16 and its numerator may be easily calculated. The denominator, however, is not so simple; it represents the standard error of the linear combination of estimated coefficients. You may recall that the variance of a sum of random variables is not merely the sum of their variances, but an expression also including their covariances, unless they are independent. The random variables {b 1, b 2, b 3 } are not independent of one another since the underlying regressors are not independent of one another. Each of the underlying regressors is assumed to be independent of the error term u, but not of the other regressors. We would expect, for instance, that firms with a larger capital stock also have a larger labor force, and use more energy in the production process. The variance (and standard error) that we need may be readily calculated by Stata, however, from the variance-covariance matrix of the estimated parameters via the test command:

17 test cap+labor+energy=1 will provide the appropriate test statistic, again as an F-statistic with a p-value. You may interpret this value directly. If you would like the point and interval estimate of the hypothesized combination, you can compute that (after a regression) with the lincom (linear combination) command: lincom cap + labor + energy will show the sum of those values and a confidence interval for that sum. We may also use this technique to test other hypotheses than adding-up conditions on the parameters. For instance, consider a two-factor Cobb-Douglas function in which you have only labor and capital, and you want to test the hypothesis that labor s share is 2/3. This implies

18 that the labor coefficient should be twice the capital coefficient, or: H 0 : β L = 2β K, or (12) H 0 : β L = 2, or β K H 0 : β L 2β K = 0 Note that this does not allow us to test a nonlinear hypothesis on the parameters: but considering that a ratio of two parameters is a constant is not a nonlinear restriction. In the latter form, we may specify it to Stata s test command as: test labor - 2*cap = 0 In fact, Stata will figure out that form if you specify the hypothesis as: test labor=2*cap

19 (rewriting it in the above form), but it is not quite smart enough to handle the ratio form. It is easy to rewrite the ratio form into one of the other forms. Either form will produce an F-statistic and associated p-value related to this single linear hypothesis on the parameters which may be used to make a judgment about the hypothesis of interest. Testing multiple linear restrictions When we use the test command, an F-statistic is reported even when the test involves only one coefficient because in general, hypothesis tests may involve more than one restriction on the population parameters. The hypotheses discussed above even that of CRTS, involving several coefficients still only represent one restriction on the parameters. For instance, if CRTS is imposed, the elasticities of the factors of production must sum to one, but they may

20 individually take on any value. But in most applications of multiple linear regression, we concern ourselves with joint tests of restrictions on the parameters. The simplest joint test is that which every regression reports: the so-called ANOVA F test, which has the null hypothesis that each of the slopes is equal to zero. Note that in a multiple regression, specifying that each slope individually equals zero is not the same thing as specifying that their sum equals zero. This ANOVA (ANalysis Of VAriance) F-test is of interest since it essentially tests whether the entire regression has any explanatory power. The null hypothesis, in this case, is that the model is y = β 0 + u : that is, none of the explanatory variables assist in explaining the variation in y. We cannot test any hypothesis on the R 2 of a regression, but we will see that

21 there is an intimate relationship between the R 2 and the ANOVA F: R 2 = SSE SST (13) F = SSE/k SSR/ (n (k + 1)) F = R 2 /k ( 1 R 2 ) / (n (k + 1)) where the ANOVA F, the ratio of mean square explained variation to mean square unexplained variation, is distributed as F n (k+1) k under the null hypothesis. For a simple regression, this statistic is Fn 2 1, which is identical to ( 2 t b1,n 2) : that is, the square of the t statistic for the slope coefficient, with precisely the same p value as that t statistic. In a multiple regression context, we do not often find an insignificant F statistic, since the null hypothesis is a very strong statement: that none of the explanatory variables, taken singly or together, explain any significant fraction of the variation

22 of y about its mean. That can happen, but it is often somewhat unlikely. The ANOVA F tests k exclusion restrictions: that all k slope coefficients are jointly zero. We may use an F-statistic to test that a number of slope coefficients are jointly equal to zero. For instance, consider a regression of 353 major league baseball players salaries (from MLB1). If we regress lsalary (log of player s salary) on years (number of years in majors), gamesyr (number of games played per year), and several variables indicating the position played ( frstbase, scndbase, shrtstop, thrdbase, catcher), we get an R 2 of , and an ANOVA F (with 7 and 345 d.f.) of with a p value of zero. The overall regression is clearly significant, and the coefficients on years and gamesyr both have the expected positive and significant coefficients. Only one of the five coefficients on the positions played, however,

23 are significantly different from zero at the 5% level: scndbase, with a negative value (-0.034) and a p value of The frstbase and shrtstop coefficients are also negative (but insignificant), while the thrdbase and catcher coefficients are positive and insignificant. Should we just remove all of these variables (except for scndbase)? The F-test for these five exclusion restrictions will provide an answer to that question:. test frstbase scndbase shrtstop thrdbase catcher ( 1) frstbase = 0.0 ( 2) scndbase = 0.0 ( 3) shrtstop = 0.0 ( 4) thrdbase = 0.0 ( 5) catcher = 0.0 F( 5, 345) = 2.37 Prob > F =

24 At the 95% level of significance, these coefficients are not each zero. That result, of course, could be largely driven by the scndbase coefficient:. test frstbase shrtstop thrdbase catcher ( 1) frstbase = 0.0 ( 2) shrtstop = 0.0 ( 3) thrdbase = 0.0 ( 4) catcher = 0.0 F( 4, 345) = 1.56 Prob > F = So perhaps it would be sensible to remove these four, which even when taken together do not explain a meaningful fraction of the variation in lsalary. But this illustrates the point of the joint hypothesis test: the result of simultaneously testing several hypotheses (that, for instance, individual coefficients are equal to zero) cannot be inferred from the results of

25 the individual tests. If each coefficient is significant, then a joint test will surely reject the joint exclusion restriction; but the converse is assuredly false. Notice that a joint test of exclusion restrictions may be easily conduced by Stata s test command, by merely listing the variables whose coefficients are presumed to be zero under the null hypothesis. The resulting test statistic is an F with as many numerator degrees of freedom as there are coefficients (or variables) in the list. It can be written in terms of the residual sums of squares (SSRs) of the unrestricted and restricted models: F = (SSR r SSR ur ) /q SSR ur / (n k 1) (14) Since adding variables to a model will never decrease SSR (nor decrease R 2 ), the restricted model in which certain coefficients are not freely

26 estimated from the data, but constrained must have SSR at least as large as the unrestricted model, in which all coefficients are data-determined at their optimal values. Thus the difference in the numerator is non-negative. If it is a large value, then the restrictions severely diminish the explanatory power of the model. The amount by which it is diminished is scaled by the number of restrictions, q, and then divided by the unrestricted model s s 2. If this ratio is a large number, then the average cost per restriction is large relative to the explanatory power of the unrestricted model, and we have evidence against the null hypothesis (that is, the F statistic will be larger than the critical point on an F table with q and (n k 1) degrees of freedom. If the ratio is smaller than the critical value, we do not reject the null hypothesis, and conclude that the restrictions are consistent with the data. In this circumstance, we might then reformulate the model

27 with the restrictions in place, since they do not conflict with the data. In the baseball player salary example, we might drop the four insignificant variables and reestimate the more parsimonious model. Testing general linear restrictions The apparatus described above is far more powerful than it might appear. We have considered individual tests involving a linear combination of the parameters (e.g. CRT S) and joint tests involving exclusion restrictions (as in the baseball players salary example). But the subset F test defined in (14) is capable of being applied to any set of linear restrictions on the parameter vector: for instance, that β 1 = 0, β 2 +β 3 +β 4 = 1, and β 5 = 1. What would this set of restrictions imply about a regression of y on {X 1, X 2, X 3, X 4, X 5 }? That regression, in its unrestricted form, would have k = 5, with 5

28 estimated slope coefficients and an intercept. The joint hypotheses expressed above would state that a restricted form of this equation would have three fewer parameters, since β 1 would be constrained to zero, β 5 to -1, and one of the coefficients {β 2, β 3, β 4 } expressed in terms of the other two. In the terminology of (14), q = 3. How would we test the hypothesis? We can readily calculate SSR ur, but what about SSR r? One approach would be to algebraically substitute the restrictions in the model, estimate that restricted model, and record its SSR r value. This can be done with any computer program that estimates a multiple regression, but it requires that you do the algebra and transform the variables accordingly. (For instance, constraining β 5 to -1 implies that you should form a new dependent variable, (y + X 5 )). Alternatively, if you are using a computer program that can test linear restrictions, you may use its features. Stata

29 will test general linear restrictions of this sort with the test command: regress y x1 x2 x3 x4 x5 test (x1) (x2+x3+x4=1) (x5=-1) This test command will print an F-statistic for the set of three linear restrictions on the regression: for instance, ( 1) years = 0.0 ( 2) frstbase + scndbase + shrtstop = 1.0 ( 3) sbases = -1.0 F( 3, 347) = Prob > F = The F-test will have three numerator degrees of freedom, because you have specified three linear hypotheses to be jointly applied to the coefficient vector. This syntax of test may

30 be used to construct any set of linear restrictions on the coefficient vector, and perform the joint test for the validity of those restrictions. The test statistic will reject the null hypothesis (that the restrictions are consistent with the data) if its value is large relative to the underlying F-distribution.

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares

Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares Many economic models involve endogeneity: that is, a theoretical relationship does not fit

### 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

### Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

### Instrumental Variables Regression. Instrumental Variables (IV) estimation is used when the model has endogenous s.

Instrumental Variables Regression Instrumental Variables (IV) estimation is used when the model has endogenous s. IV can thus be used to address the following important threats to internal validity: Omitted

### The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information

Chapter 8 The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information An important new development that we encounter in this chapter is using the F- distribution to simultaneously

### Wooldridge, Introductory Econometrics, 4th ed. Chapter 7: Multiple regression analysis with qualitative information: Binary (or dummy) variables

Wooldridge, Introductory Econometrics, 4th ed. Chapter 7: Multiple regression analysis with qualitative information: Binary (or dummy) variables We often consider relationships between observed outcomes

### DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests

DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also

### 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

### Chapter 5: Basic Statistics and Hypothesis Testing

Chapter 5: Basic Statistics and Hypothesis Testing In this chapter: 1. Viewing the t-value from an OLS regression (UE 5.2.1) 2. Calculating critical t-values and applying the decision rule (UE 5.2.2) 3.

### 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

### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

### 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

### Solución del Examen Tipo: 1

Solución del Examen Tipo: 1 Universidad Carlos III de Madrid ECONOMETRICS Academic year 2009/10 FINAL EXAM May 17, 2010 DURATION: 2 HOURS 1. Assume that model (III) verifies the assumptions of the classical

### 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

### Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### Regression, least squares

Regression, least squares Joe Felsenstein Department of Genome Sciences and Department of Biology Regression, least squares p.1/24 Fitting a straight line X Two distinct cases: The X values are chosen

### 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.

### SIMPLE REGRESSION ANALYSIS

SIMPLE REGRESSION ANALYSIS Introduction. Regression analysis is used when two or more variables are thought to be systematically connected by a linear relationship. In simple regression, we have only two

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### Regression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.

Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

### 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

### Statistics 112 Regression Cheatsheet Section 1B - Ryan Rosario

Statistics 112 Regression Cheatsheet Section 1B - Ryan Rosario I have found that the best way to practice regression is by brute force That is, given nothing but a dataset and your mind, compute everything

### 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

### Lecture 5 Hypothesis Testing in Multiple Linear Regression

Lecture 5 Hypothesis Testing in Multiple Linear Regression BIOST 515 January 20, 2004 Types of tests 1 Overall test Test for addition of a single variable Test for addition of a group of variables Overall

### Instrumental Variables & 2SLS

Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### 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

### Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

### Instrumental Variables & 2SLS

Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### Factorial Analysis of Variance

Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

### Statistical Functions in Excel

Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

### Wooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions

Wooldridge, Introductory Econometrics, 3d ed. Chapter 12: Serial correlation and heteroskedasticity in time series regressions What will happen if we violate the assumption that the errors are not serially

### Chapter 9. Section Correlation

Chapter 9 Section 9.1 - Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

### Regression step-by-step using Microsoft Excel

Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression

### 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

### Regression analysis in practice with GRETL

Regression analysis in practice with GRETL Prerequisites You will need the GNU econometrics software GRETL installed on your computer (http://gretl.sourceforge.net/), together with the sample files that

### MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL. by Michael L. Orlov Chemistry Department, Oregon State University (1996)

MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL by Michael L. Orlov Chemistry Department, Oregon State University (1996) INTRODUCTION In modern science, regression analysis is a necessary part

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Linear Models in STATA and ANOVA

Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 4-2 A Note on Non-Linear Relationships 4-4 Multiple Linear Regression 4-5 Removal of Variables 4-8 Independent Samples

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### Mgmt 469. Regression Basics. You have all had some training in statistics and regression analysis. Still, it is useful to review

Mgmt 469 Regression Basics You have all had some training in statistics and regression analysis. Still, it is useful to review some basic stuff. In this note I cover the following material: What is a regression

### 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,

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

### 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

### " 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

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

### 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.

### Review of Bivariate Regression

Review of Bivariate Regression A.Colin Cameron Department of Economics University of California - Davis accameron@ucdavis.edu October 27, 2006 Abstract This provides a review of material covered in an

### What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling

What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and

### Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

### One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups In analysis of variance, the main research question is whether the sample means are from different populations. The

### ECON 523 Applied Econometrics I /Masters Level American University, Spring 2008. Description of the course

ECON 523 Applied Econometrics I /Masters Level American University, Spring 2008 Instructor: Maria Heracleous Lectures: M 8:10-10:40 p.m. WARD 202 Office: 221 Roper Phone: 202-885-3758 Office Hours: M W

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### ABSORBENCY OF PAPER TOWELS

ABSORBENCY OF PAPER TOWELS 15. Brief Version of the Case Study 15.1 Problem Formulation 15.2 Selection of Factors 15.3 Obtaining Random Samples of Paper Towels 15.4 How will the Absorbency be measured?

### individualdifferences

1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### Lecture 2: Simple Linear Regression

DMBA: Statistics Lecture 2: Simple Linear Regression Least Squares, SLR properties, Inference, and Forecasting Carlos Carvalho The University of Texas McCombs School of Business mccombs.utexas.edu/faculty/carlos.carvalho/teaching

### An Introduction to Regression Analysis

The Inaugural Coase Lecture An Introduction to Regression Analysis Alan O. Sykes * Regression analysis is a statistical tool for the investigation of relationships between variables. Usually, the investigator

### Testing for serial correlation in linear panel-data models

The Stata Journal (2003) 3, Number 2, pp. 168 177 Testing for serial correlation in linear panel-data models David M. Drukker Stata Corporation Abstract. Because serial correlation in linear panel-data

### Practical. I conometrics. data collection, analysis, and application. Christiana E. Hilmer. Michael J. Hilmer San Diego State University

Practical I conometrics data collection, analysis, and application Christiana E. Hilmer Michael J. Hilmer San Diego State University Mi Table of Contents PART ONE THE BASICS 1 Chapter 1 An Introduction

### International Statistical Institute, 56th Session, 2007: Phil Everson

Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction

### Chapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS

Chapter Seven Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Section : An introduction to multiple regression WHAT IS MULTIPLE REGRESSION? Multiple

### General Procedure for Hypothesis Test. Five types of statistical analysis. 1. Formulate H 1 and H 0. General Procedure for Hypothesis Test

Five types of statistical analysis General Procedure for Hypothesis Test Descriptive Inferential Differences Associative Predictive What are the characteristics of the respondents? What are the characteristics

### 1 Another method of estimation: least squares

1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

### Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1. 1. Introduction p. 2. 2. Statistical Methods Used p. 5. 3. 10 and under Males p.

Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1 Table of Contents 1. Introduction p. 2 2. Statistical Methods Used p. 5 3. 10 and under Males p. 8 4. 11 and up Males p. 10 5. 10 and under

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

Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

### DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

### Chapter 8 Introduction to Hypothesis Testing

Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### 2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or

Simple and Multiple Regression Analysis Example: Explore the relationships among Month, Adv.\$ and Sales \$: 1. Prepare a scatter plot of these data. The scatter plots for Adv.\$ versus Sales, and Month versus

### 2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### STAT 350 Practice Final Exam Solution (Spring 2015)

PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

### Develop hypothesis and then research to find out if it is true. Derived from theory or primary question/research questions

Chapter 12 Hypothesis Testing Learning Objectives Examine the process of hypothesis testing Evaluate research and null hypothesis Determine one- or two-tailed tests Understand obtained values, significance,

### EPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM

EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable

### e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

### 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) -

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the