Blog

Topic 1 The Basic Linear Model 1. Assumptions in Linear Regression (1) Assumptions about the model – MR1: The correct model is: = 0 + 11 + 22 + + + (2) Assumptions about the error term: – MR2: Zero Conditional Expectation: The conditional expected value of the random errors is zero: (|) = 0 – MR3: Homoskedasticity: The variance of the random errors is constant and independent of : (|) = 2 – MR4: No Autocorrelation: Any pair of random errors are uncorrelated: (, |, ) = 0 , = 1,2, …, ≠ – MR0: Not a necessary assumption: Normally Distributed Error Term | ~ (0, 2) If you assume the error term is normally distributed, then you can write the model as: = 0 + 11 + 22 + + + | ~ (0, 2), or | ~ (0 + 11 + 22 + + , 2) (3) Assumptions about the Explanatory Variables: = (1, 2, … , ) – MR5a: The explanatory variables satisfy any one of the following: Non-stochastic Regressor: The explanatory variables are not random Exogeneity: The explanatory variables are random and uncorrelated with the error term (, ) = 0 , = 1,2, … – MR5b: No Exact Collinearity / No Perfect Multicollinearity: Any one of the X’s is not an exact linear function of any of the other X’s. 2. The Gauss-Markov Theorem – Under the assumptions of the linear regression model (MR1 – MR5), the OLS estimators = ( 0, 1, 2,… , ) are the Best Linear Unbiased Estimators (BLUE) (have the smallest variance) of all linear and unbiased estimators of = (0, 1, 2,… , ) Topic 2 Statistical Properties & Inferences in Linear Model Unit 1 Sampling Distribution of OLS Estimators 1. Sampling Distribution of OLS estimators – If all MR1 – MR5 satisfied, the sampling distribution of OLS estimators: ~(,) ( 0 1 2 ) ~ ( ( 0 1 2 ) , ( (0) (0, 1) (0, 2) (0, ) (0, 1) (1) (1, 2) (1, ) (0, 2) (1, 2) (2) (2, ) (0, ) (1, ) (2, ) () ) ) 2. Mean of OLS Estimators & Unbiasedness – The mean of OLS estimators: (|) = for = 1,2, … if (|) = 0 for = 1,2, … (MAR2 holds) – OLS estimators are unbiased estimators for population parameters, since (|) is equal to , which is the parameter that is going to estimate 3. Variance of OLS Estimators (1) Variance-Covariance Matrix – The variance-covariance matrix of OLS estimators will be correct if all the following holds (|) = 0 for = 1,2, … (MAR2 holds) (|) = 2 (MR3 holds) (, |, ) = 0 for all , = 1,2, …, ≠ (MR4 holds) The ′ are non-random (MR5a holds) – If all the conditions above hold, the formula for variance of OLS estimators: () = 2 ∑( ) 2 = 2 ( 1)() (2) Factors Affecting the Variance of the OLS Estimators (a) Variation of Residuals (Positively Affected): 2 (b) Variation in Explanatory Variables (Negatively Affected): ∑( ) 2 (c) Sample Size (Negatively Affected): (d) Correlation between Explanatory Variables (Positively Affected): (, ) Unit 2 Measuring Goodness of Fit in Linear Model 1. ANOVA Table in Regression & Estimator of Error Variance 2 Sum Square d.f. Mean Square F Statistic for Overall Significance Model =∑( ) 2 =1 = / = / /( 1) Residual =∑( ) 2 =1 1 = /( 1) Total =∑( ) 2 =1 1 – The estimators of error variance: 2 = ∑ 2 =1 ( 1) = ∑ ( ) 2 =1 ( 1) = ( 1) = where : number of explanatory variables / + 1: number of parameters OLS residuals = ( ) = (0 + 11 + 22 + + ) 2. Measuring Goodness of Fit in Linear Model (1) Coefficient of Determination: 2 2 = 1 ∑ 2 =1 ∑ ( )2 =1 = 1 = (, ) – Coefficients of determination: 2 is the percentage of the variation in the dependent variable about its mean that is explained by the regression model – Problem of 2: 2 cannot decrease when including irrelevant variables in the model (2) Adjusted 2: 2 2 = 1 × 1 1 = 1 [(1 2) × 1 1 ] – Interpretation of 2: 2 does not measure the percentage of the variation in the dependent variable explained by the regression model 2 can be negative Unit 3: Hypothesis Testing 1. OLS t Test – Test statistic: = () ~( 1) where: + 1: number of explanatory variables in the model : value of stated in 0 : OLS estimator for (): Standard error of , which is the square root of () – Rejection Rule: 0 Critical Value Method Reject 0 if p-value Method Reject 0 if < 0: = : ≠ < (1 2 , 1) > (1 2 , 1 = 2 × Pr( > ) for > 0 = 2 × Pr( < ) for < 0 : < < (, 1) = Pr( < ) : > > (1 , 1) = Pr( > ) 2. Type I Error and Type II Error Reject 0 Not Reject 0 When 0 is True Type I Error Pr( 0| 0 ) = Significance Level / Size of Test Correct Decision Pr( 0| 0 ) = 1 Confidence Level When 0 is False Correct Decision Pr( 0| 0 ) = 1 Power of Test Type II Error Pr( 0| 0 ) = – Type I error is the significance level we set – Type II error is not under our control Probability of Type II error is unknown because the true parameter value is unknown Probability of Type II error can be reduced by larger sample size – Probability of Type I error & Type II error is negatively related Probability of Type II error can be reduced by setting a larger 3. The General OLS F Test – The F Test Statistic = ( )/ /( 1) ~(, 1) where: : Residual Sum of Square in restricted model : Residual Sum of Square in unrestricted model : Number of restrictions + 1: Number of parameters – The Rejection Region Reject 0 if > (1 ,, 1), or Reject 0 if = Pr( > ) < 4. F Test of Significance of Model - The F Test Statistic = ( )/ /( 1) = ( )/ /( 1) = (2)/ (1 2)/( 1) ~( = , 1) where: : Residual Sum of Square in restricted model / : Residual Sum of Square in unrestricted model : Total Sum of Square in unrestricted model : Number of restrictions + 1: Number of parameters 2: Coefficient of Determination of unrestricted model - The Rejection Region Reject 0 if > (1 , = , 1) Reject 0 if = Pr( > ) < Topic 3 Model Specification Review MR1: Model Specification: The correct model is: = 0 + 11 + 22 + + + The functional form selection is correct No omission of relevant explanatory variables and No inclusion of irrelevant explanatory variables Unit 1 Functional Form Selection - Four Main Types of Functional Form: Functional Form Dependent Variable Explanatory Variable Interpretation of Slop Coefficient Linear-Linear Model = 0 + 1 + Unit form Unit form One unit increase in is associated with 1 unit increase in Log-Linear Model log = 0 + 1 + Logarithmic form Unit form One unit increase in is associated with 1 × 100% percentage increase in Linear-Log Model = 0 + 1 log + Unit form Logarithmic form 100% percentage increase in is associated with 1 unit increase in Log-Log Model log = 0 + 1 log + Logarithmic form Logarithmic form 1% percentage increase in is associated with 1% percentage increase in (Elasticity) - Additional Types of Functional Forms (1) Polynomial = 0 + 1 + 2 2 + 3 3 + = 0 + 11 + 22 + 312 + 41 2 + 52 2 + (2) Reciprocal = 0 + 1 1 + Unit 2 Omission of Relevant Explanatory Variables - Consider the correct model is: = 0 + 1 + 2 + and the incorrect model, which omit , is: = 0 + 1 + with = 2 + - The OLS estimation of the model above, which omit : (1) = 1 + 2 (, ) () What we want to estimate is 1 The actual estimate is: 1 + 2 (,) () The difference: 2 (,) () is the biased amount Unit 3 Inclusion of Irrelevant Explanatory Variables - Consider the correct model is: = 0 + 1 + and the incorrect model, which included an irrelevant , is: = 0 + 1 + 2 + - Inclusion of irrelevance explanatory variable will cause: The OLS estimators of the parameters will be unbiased, because is an irrelevant explanatory variable, which implies 2 = 0 The variances of the OLS estimators in the model will be larger Unit 4 Testing for Model Specification 1. RESET Test for Model Specification - Consider the following model with fitted model: = 0 + 11 + 22 + = 0 + 11 + 22 - The RESET test model: Test reasonableness to include binomial terms: = 0 + 11 + 22 + 1 2 + with Hypothesis: 0: 1 = 0 against : 1 ≠ 0 Test reasonableness to include binomial terms: = 0 + 11 + 22 + 1 2 + 2 3 + with Hypothesis: 0: 1 = 2 = 0 against : 1 ≠ 0 2 ≠ 0 - Conclusion: Reject 0 implies that original model specification is not correct Not reject 0 implies that original model specification is correct - Including powers of 2 can test: Incorrect functional form: Consider the reasonableness to include polynomial terms Omitted variable: Including polynomial terms may reflect omitted variables related to included explanatory variables Note: RESET test cannot identify exact reason for model misspecification: functional form, or omitted variables, or combination of two sources 2. Jarque-Bera Test for Normality of Error Term - Null hypothesis and alternative hypothesis: 0: Errors are normally distributed : Errors are not normally distributed - Test Statistic: = 6 [2 + ( 3)2 4 ]~2( = 2) where: : Sample skewness of OLS residuals : Sample kurtosis of OLS residuals : Sample size - Conclusion of Jarque-Bera Test Reject 0: errors are not normally distributed and model specification is incorrect Not reject 0: errors are normally distributed and model specification is correct Unit 5: Collinear Economic Variables Review MR5b: No Exact Collinearity: Any one of the X’s is not an exact linear function of any of the other X’s. Exact collinearity causes OLS estimation to be invalid Near-exact collinearity causes OLS estimation valid, but cause problems in estimation 1. Introduction to Collinearity - Collinearity is the problem that two or more explanatory variables are highly correlated - Consequence of collinearity (1) OLS estimators are still unbiased, but variances and covariances of the OLS estimators might be very large (2) The individual coefficient t test statistic is not significance, but the model fitness is still good, with high 2 and significant F test statistics (3) Estimates may be sensitive to addition deletion of explanatory variables - Solution to collinearity Exclude explanatory variables until no collinearities among explanatory variables However, there may be trade-off between omitted variables and collinearity 2. Test for Collinearity (1) Pairwise Correlation - Test the sample correlation between each pair of explanatory variables: - For example, consider the model: = 0 + 11 + 22 + 33 + The pairwise correlations: (1, 2) & (1, 3) & (2, 3) If there is correlation that is significantly different from 0, there should be collinearity between suggested explanatory variables (2) Auxiliary Regressions - Construct regressions with one original explanatory variable to be new dependent variable, and all the other original explanatory variables to be new explanatory variables - For example, consider the model: = 0 + 11 + 22 + 33 + The auxiliary regressions 1 = 0 + 12 + 23 + 2 = 0 + 11 + 23 + 3 = 0 + 11 + 22 + If the 2 is high in one or more equations above, there should be collinearity among original explanatory variables Topic 5 Heteroskedasticity Unit 1 Nature of Heteroskedasticity 1. Definition of Heteroskedasticity - Heteroskedasticity: violation of MR3: The variance of error depends on data: (|) = 2 Variance of the error may depend on explanatory variables: Variance of the error may depend on other factors of data - Consequences of heteroscedasticity (1) The OLS estimator is still a linear and unbiased estimator but not BLUE (2) The standard errors of OLS estimator are incorrect (3) Hypothesis tests & confidence intervals will be misleading - Solution to Heteroskedasticity: Use robust / White standard errors Use generalised least squares Unit 2 Generalised Least Squares - If we know the form of the heteroskedasticity, we can use generalised least squares - In general, if we know that (|) = 2 = 2, we can transform the model = 0 + 1 + √ = 0 √ + 1 √ + √ = 01 + 12 + - New error term satisfies all the assumptions ( ) = ( √ ) = 1 √ () = 1 √ × 0 = 0 ( ) = ( √ ) = ( 1 √ ) 2 () = 1 2 = 2 ( , ) = ( √ , √ ) = 1 √ × 1 √ (, ) = 1 √ × 1 √ × 0 = 0 - The estimator for transformed model is called GLS Estimators, and is BLUE Unit 3 Feasible Generalised Least Squares 1. Feasible Generalised Least Squares (FGLS) - FGLS can solve heteroskedasticity with unknown form - Consider the following model: = 0 + 11 + 22…+ + we know is heteroskedasticity but we do not know the form - The function form of heteroskedasticity by FGLS: (|) = 2 × exp{0 + 11 + 22 …+ } = 2 ′ are unknown parameters constructing the weight for GLS Use exponential for to ensure positive variance homoskedasticity when 1 = 2 = = 2. Forms of FGLS - Transformation of the model: 2 = 2 × exp{0 + 11 + 22…+ } ln 2 = ln2 + 0 + 11 + 22…+ ln 2 = ln 2 + 0 + 11 + 22…+ + ln 2 ln 2 ln 2 = 0 + 11 + 22…+ + with 0 = 0 + ln 2 and = ln 2 ln 2 = ln ( 2 2) = ln ( ) 2 but ~ ln 2(1) (|) = 1.2707 ≠ 0 - Further Transformation ln 2 = 0 + 11 + 22…+ + ln 2 = 0 1.2707 + 11 + 22…+ + + 1.2707 = ln 2 = 0 + 11 + 22…+ + with 0 = 0 1.2707 and = + 1.2707 here ( |) = 0 3. Steps in FGLS (1) Estimate the original model and get residuals: = = 0 + 11 + 22…+ + (2) Use residuals to estimate: = ln 2 = 0 + 11 + 22…+ + Note: Estimated intercept will be off by -1.2707 Estimated slope will be consistent (valid in large sample) (3) Use fitted value of to run GLS: √exp( ) = 0 √exp( ) + 11 √exp( ) + 22 √exp( ) …+ √exp( ) + √exp( ) with 2 = (|) = 2 × exp() ( 1 √exp( ) ) = ( 1 √exp( ) ) 2 () = 2 Unit 4 Detecting Heteroskedasticity 1. Plot Method (1) Plot of against with fitted line (2) Plot of against residuals 2. White’s Test - White’s test can test heteroskedasticity with unknown form by comparing White’s standard error & OL

S standard error – Hypothesis: 0: 2 = 2 (Homo) against : 2 ≠ 2 (Homo) against with 2: White’s standard error 2: OLS standard error – Test Statistics If the model is: = 0 + 11 + 22 + The aux regression of squared residuals on all possible combinations of ′ 2 = 0 + 11 + 22 + 31 2 + 42 2 + 512 + The test statistics: × 2~2( 1) with : Sample Size 2: 2 of aux regression : Number of parameters in aux regression – Problem of White’s test White’s test is in fact a test for model specification 0 assume that Error is homoskedasticity Error in independent of ′ Functional form is correct Reject 0 implies that one or more conditions above not satisfied, which may or may not identify heteroskedasticity Topic 6 Time Series Unit 1 Autocorrelation 1. The Nature of Autocorrelation – The problem of autocorrelation in error term: Violation of MR4: Error terms are correlated over-time (, ) ≠ 0 for some – Consequence of autocorrelation (4) The OLS estimator is still a linear and unbiased estimator but not BLUE (5) The standard errors of OLS estimator are incorrect (6) Hypothesis testing & confidence interval are invalid – Solution to autocorrelation: Use HAC standard errors Use generalised least squares: with known form of autocorrelation 2. First-Order Autoregressive Process in Error Term: (1) – If we know that the error term is autocorrelated one-period-ahead (, 1) ≠ 0 we can express its form in (1) time-series regression: = 1 + – The error term is composed of two parts: 1: The carry-over from previous period : The correlation in 1 < < 1: The carry-over will diminish period by period : The new shock this is uncorrelated overtime, and satisfies Gauss-Markov Theorem - Properties of First-Order Autoregressive Errors: Zero mean () = 0 Homoscedasticity () = 2 = 2 1 2 Covariance between successive error terms: (, ) = 2 = 2 1 2 Correlation between successive error terms: (, ) = (, ) √()√( ) = 2 √2√2 = 3. Generalised Least Square (1) Methods of Generalised Least Square - The time series regression model: = 0 + 1 + with = 1 + with properties () = 0 () = 2 = 2 1 2 (, ) = 2 = 2 1 2 - Substitute the (1) error into the model: = 0 + 1 + = 0 + 1 + ( 1 + ) 1 = 1 0 1 1 1 = 0 + 1 1 + 1 = 0 + 1 + ( 1 0 1 1) + 1 = 0(1 ) + 1( 1) + - The transformed model: = 0 + 1 + with = 1; 0 = 0(1 ); 1 = 1; = 1 where in the new model satisfies Gauss-Markov Theorem (T-1) observations are included in new model (2) Steps in Generalised Least Square Step 1: Estimate by OLS - Since = 1 + , we ca estimate using OLS residuals as estimation of error term Obtain the OLS residuals = 0 1 Put OLS residuals into (1) model = 1 + Estimation of = ∑ 1 =2 ∑ 1 2 =2 Step 2: Implement GLS - The transformed model: = 0 + 1 + with = 1; 0 = 0(1 ); = 1 4. Test for Autocorrelation – LM Test - Consider the following time series model: = 0 + 1 + where we want to test whether: (, ) ≠ 0 for some ≥ 1 - LM Test for (1) errors: = 1 + Hypothesis: 0: = 0 against 1: ≠ 0 Test Statistics: Using 2 of the following model = 0 + 1 + 1 + = × 2~2(1) Reject 0 if > 2(, 1) – LM Test for (2) errors: = 1 1 + 2 2 + Hypothesis: 0: 1 = 2 = 0 against 1: At least one ≠ 0, = 1,2 Test Statistics: Using 2 of the following model = 0 + 1 + 1 1 + 2 2 + = × 2~2(2) Reject 0 if > 2(, 2) – Test for () errors: = 1 1 + 2 2 + + + Hypothesis: 0: 1 = 2 = = = 0 against 1: At least one ≠ 0, = 1,2… Test Statistics: Using 2 of the following model = 0 + 1 + 1 1 + 2 2 + + + = × 2~2() Reject 0 if > 2(, ) – Problem of LM Test LM tests need us to know the form of autocorrelation If we do not know the form of autocorrelation: Testing Down Approach Testing up a pp roach Unit 2 Non-Stationary Time Series – In time series models, all time series data included should be stationary – Summary: Time Series Patterns Name Plot Stationarity Unit Root ADF Test Type Stationary Time Series / Mean- Reverting Stationary No Drift Trend Stationary Non-Stationary No Drift Random Walk Non-Stationary Yes Trend Random Walk with Drift Non-Stationary Yes Trend 1. Stationary Time Series (1) Definition of Stationary – A time series data is stationary if: Its mean is constant overtime () = Its variance is constant overtime () = 2 Its autocovariance for any given lag length is constant overtime (, ) = (, +) (2) Stationary Time Series in (1) Model – The (1) model of which is stationary: = 1 + with || < 1 where satisfies Gauss-Markov Theory - Then, we have: () = 0 () = 2 ( 1 1 2 ) (, +) = 2 ( 1 2 ) (3) Different Kinds of (1) Model with Stationary Time Series - (1) Model with zero mean = 1 + - (1) Model with non-zero mean = + 1 + with () = = 1 De-meaning: = + 1 + = (1 ) + 1 + = ( 1 ) + with = (1 ) - (1) Model with non-zero mean & linear trend (Trend Stationary which is Non-Stationary) = + + 1 + with () = + De-meaning & de-treading: = + + 1 + = [(1 ) + ] + (1 ) = [ 1 ( 1)] + with = (1 ) + and = (1 ) 2. Non-Stationary Time Series - In the (1) model of , if = 1, then is non-stationary (1) Random Walk Model = 1 + Apply recursive substitution = 0 +∑ 1 =0 then () = (0) = 0 () = (0 +∑ 1 =0 ) = 2 Random walk is non-stationary because its variance increases over time (2) Random Walk with Drift = + 1 + Apply recursive substitution = + 0 +∑ 1 =0 then () = + 0 () = 2 Random walk is non-stationary because its mean and variance increases over time 3. Spurious Regression - Definition of Spurious Regression If we regress two non-stationary time series on each other, we will get a significant OLS coefficient even if the series are unrelated Consider the following two random walk time series = 1 + 1 = 1 + 2 and are uncorrelated, however in the following model: = 0 + 1 + estimate of 1 is usually statistically significant suggesting and are correlated - Consequence of Spurious Regression OLS estimators’ statistical properties are not valid OLS estimators’ statistical inferences (hypothesis testing) are not correct 4. Testing for Non-Stationarity: Dickey Fuller (DF) Test (1) Dickey Fuller (DF) Test - Consider the model: = 1 + or Δ = ( 1) 1 + = 1 + - Hypothesis: 0: Non-Stationary / = 1 / = 0 0: Stationary / < 1 / < 0 - Different Kinds of Models DF Test I: If time series appears to fluctuate around mean zero: Δ = 1 + with 0: = 0 DF Test II: If time series appears to fluctuate around non-zero mean: Δ = + 1 + with 0: = 0 DF Test III: If time series appears to fluctuate around linear trend: Δ = + + 1 + with 0: = 0 (2) Augmented Dickey Fuller (ADF) Test - ADF test consider possible autocorrelation in error term - We add augmented terms (lagged first difference terms in ) to eliminate autocorrelation Δ = + 1 + ∑ Δ =1 + with 0: = 0 (3) Test Statistics - The test statistics for DF test & ADF test follows DF distribution, not t distribution - We just focus on p-value of the tests Topic 7 Stochastic Regressors Unit 1: Stochastic Regressors and Problem of Endogeneity 1. Definition of Endogeneity - MR5a version 2: Exogeneity requires that the explanatory variables are random and uncorrelated with the error term: ( , ) = 0 - Endogeneity is a problem that explanatory variable is correlated with error term: ( , ) ≠ 0 2. Consequence of Endogeneity - OLS estimators are biased and inconsistent - OLS standard errors are incorrect - Hypothesis tests & confidence interval are invalid 3. Sources of Endogeneity (1) Classical Measurement Error (2) Omitted Variables (3) Lagged Dependent Variable with (1) Error Structure (4) Simultaneous Causal Equations Unit 2: Sources of Endogeneity 1. Classical Measurement Error - Classical measurement error arises when explanatory variables are measured with error - Consequences of Classical Measurement Error OLS estimators are biased, and standard errors are incorrect Attenuation Bias: 1 is biased towards zero No bias if and only if no measurement error 2. Omitted Variables - Omitted variable bias arise if One or more relevant explanatory variables are omitted from the true mode, and Omitted explanatory variables are correlated with included explanatory variables - Consider the correct model is: = 0 + 1 + 2 + with (,) = 0 (, ) = 0 - Consider the incorrect model, which omit : = 0 + 1 + (2 + ) = 0 + 1 + - In this estimated model: (, ) = (, 2 + ) = 2(,) + ( , ) = 2(,) Endogeneity arises if 2 ≠ 0 and (,) ≠ 0 - The incorrect OLS estimation: (1) = 1 + 2 (,) () The bias: = 2 (,) () 3. Lagged Dependent Variables - In time series model, when including a lag of dependent variable as an explanatory variable: = 0 + 1 1 + - If there is autocorrelation in error term correlated with 1 1 correlated with 1: endogeneity Unit 3: Instrumental Variable (IV) and Two Stage Least Squares (2SLS) - Instrumental Variable (IV) and Two Stage Least Squares (2SLS) is the effective and efficient method to solve all sources of endogeneity 1. Instrumental Variable Validity Condition - Consider the model where is endogenous = 0 + 1 + - Consider the instrumental variable , that satisfies IV validity conditions (a) Exclusive Restriction does not directly correlated with dependent variable does not correlated with error term: (, ) = 0 (b) Instrumental Relevance is correlated with endogenous variable ( , ) ≠ 0 2. Two Stage Least Square - Consider the model where is endogenous and is a valid IV for = 0 + 1 + - The process of Two Stage Least Squares (2SLS) estimation: (a) Stage 1: Regress on to get a fitted value of : = 0 + 1 (b) Stage 2: Estimate original model, replacing with fitted value of = 0 + 1 - By the process of 2SLS, the estimator 1 is called 2SLS estimator or IV estimator 3. Properties of IV Estimators (1) IV estimators are consistent if IV validity conditions satisfied (2) In small sample, IV estimators are biased (3) In large sample, IV estimators are approximately unbiased and normally distributed - In the model: = 0 + 1 + where is endogenous and is a valid IV for - The sampling distribution of IV estimator in large sample 1 . ~ (1, 2 2 2 ) (4) The Variance of IV Estimator - In large sample, IV estimators have variance: (1 ) = 2 2 2 where 2: Population variance of 2: Population variance of 2 : Square of population correlation between and - Estimates of Variance of IV Estimator (1 ) = 2 2 2 = 2 ∑( )2 2 where 2 : Sample variance of residuals 2: Sample variance of 2 : Square of sample correlation between and - Properties of Variance of IV Estimator (a) Suppose (, ) = 0, then for the OLS estimator & IV estimator (1 ) = 2 ∑( )2 = 2 (1 ) When OLS is valid, OLS estimator will have a lower variance than IV estimator (b) Higher sample correlation 2 lowers the estimated variance of IV estimator (c) Larger sample size lower variance of IV estimator (d) Greater variance in lower variance of IV estimator 4. Weak Instruments - Weak Instrument: (, ) ≈ 0 - How to detect weak instrument: Rule of thumb: F test statistic of IV in stage 1 less than 10 - Problem of weak instrument: Normal distribution provides poor approximation to sampling distribution of the IV estimator in large sample No justification for using t-tests and F-tests which are based on assumption of normality Statistical inference: affects conclusions from hypothesis tests, which are based upon an incorrect distribution of the test statistic when 0 is true Unit 4: Simultaneous Causal Equation System 1. Definition of Simultaneous Causal Equations - Simultaneous Causal Equations: Two variables are jointly determined - Simultaneous causal equations is one source of endogeneity: (, ) 2. Simultaneous Causal Equations: Structural Equations - Consider two simultaneous caused variables: 1 & 2 { 1 = 12 + 11 + 1 (1) 2 = 21 + 22 + 2 (2) (1) Endogenous Variables - In the equations system above, endogenous variables: 2 is the endogenous variable in equation 1 1 is the endogenous variable in equation 2 1 & 2 are both the endogenous variables in the system (2) Exogenous Variables & Instrumental Variables - In simultaneous causal equation system, all exogenous variables in the system can be considered as instrumental variables, irrespective of which equation they are in - In the equations system above, exogenous variables: 1 & 2 are the exogenous variables / instrumental variables in the system 1 is the exogenous variable in only equation 1 1 is used to identify equation 1 1 can be used as exclusive IV for 1 2 is the exogenous variable in only equation 2 2 is used to identify equation 2 2 can be used as exclusive IV for 2 (3) Identification - In simultaneous equation system, an equation is identified if it contains exogenous variable that does not exist in other equations - In a system of 2 simultaneous equations with 2 endogenous variables, at least 1 equations must be identified; we need at least 1 exogenous variables - If a simultaneous equation system is identified, the parameters in this system can be estimated consistently 3. Simultaneous Causal Equations: Reduced Forms (1) Definition and Illustration of Reduced Forms - The reduced forms express each endogenous variable in terms of all exogenous variables in the system - Consider the structural equations: { 1 = 12 + 11 + 1 (1) 2 = 21 + 22 + 2 (2) assuming: (1) = (2) = 0 (1, 2) = 0 - Reduced forms: { 1 = 111 + 122 + 1 (1) 2 = 211 + 222 + 2 (2) where coefficients ′ are functions of structural coefficients (2) Properties of Reduced Form - Reduced form can be estimated by OLS, because no endogeneity in re