ECON 104 W24 - Midterm Review

Heteroskedasticity Time Series Endogeneity Simultaneous Equations What to study The "style" of questions from the practice/previous midterm on the website, however the way "coding" questions are implemented will be different. In particular, questions 5, 9, 12, and 13 in the practice midterm There will still be problems that require you to interpret code, and not just code output Labs (including that Econ 103 lab PDF) are the best tools to study the necessary code. Reading/understanding code output is still required! The discussion problems are also useful to study.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Heteroskedasticity Overview How do we test for it? 1 Breusch-Pagan (BP) Test White Test 2 Goldfield-Quandt (GQ) Test How do we fix it? 1 White Standard Errors (HC) 2 Weighted Least Squares

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Goldfield-Quandt (GQ) Test The GQ test is a test on GROUPS ; We want to see if σ 2 1 = σ 2 2 1 Arrange the observations in ascending order of the variable you believe is the source of heteroskedasticity (Skip if discrete) 2 Split into two groups 3 Run two separate regressions and compute their variance: ˆ σ 2 g = SSE g N g − K g 4 Compute the test statistic (with σ 1 > σ 2 ): ˆ GQ = ˆ σ 2 1 ˆ σ 2 2 ∼ F ( N 1 − K 1 , N 2 − K 2 ) 5 If ˆ GQ > F α ( N 1 − K 1 , N 2 − K 2 ) , then heteroskedasticity is present

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Weighted Least Squares What affect does this have on the variance of our residuals? Var( e ∗ i | x i ) = Var e i p h ( x i ) x i ! = 1 h ( x i ) Var( e i | x i ) = 1 h ( x i ) σ 2 h ( x i ) = σ 2 After our weighting, we have a homoskedastic model! → Our parameters β 1 and β 2 are unchanged → The standard errors of the model are changed: Var( b 2 | X ) WLS = N N − K ∑ N i = 1 [( x ∗ i − ¯ x ∗ ) 2 ˆ σ 2 ] ∑ n i = 1 ( x ∗ i − ¯ x ∗ ) 2 2 (9)

Heteroskedasticity Time Series Endogeneity Simultaneous Equations WLS vs GLS WLS assumes you know the function h ( x ) , GLS estimates this function: WLS might have specification error GLS might have estimation error With error, some heteroskedasticity may remain. −→ Add on a White standard error and you are good! The residual variances for the two methods differ: WLS has Var(ˆ e 2 i ) = σ 2 GLS has Var(ˆ e 2 i ) = 1 This happens because the intercept term in the GLS function θ 0 corresponds to σ 2 : ˆ w i = ˆ σ 2 ˆ h ( x i )

Heteroskedasticity Time Series Endogeneity Simultaneous Equations 3 Types of Time Series Models Autoregressive: Relates previous values of Y with current outcomes of Y : Y t = β + β 1 Y t − 1 + · · · + β p Y t − p + e t (AR(p)) Distributed Lag: Relates previous values of X with current outcomes of Y : Y t = δ + δ 0 X t + δ 1 X t − 1 + · · · + δ q X t − q + e t (DL(q)) ARDL: Combines the previous two models into one: Y t = θ + β 1 Y t − 1 + · · · + β p Y t − p + δ 0 X t + δ 1 X t − 1 + · · · + δ q X t − q + e t (ARDL(p,q))

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Correlograms The correlogram is a representation of sample correlations at various lengths. A horizonal line is drawn at ± 2 / √ T , which is approximately the critical value:

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Partial Autocorrelations Idea is to "partial out" the intermediate lags of the ACF to isolate the lag of interest.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Breusch-Godfrey (BG) Test A Breusch-Godfrey test is a regression of today’s errors on past errors. Suppose you had the following ARDL model: y t = δ + θ 1 y t − 1 + δ 1 x t − 1 + e t (17) and wanted to know if the errors had autocorrelation at some lag s ? Solution: Estimate the equation, get the residuals ˆ e t , and run the following regression: ˆ e t = δ + ρ 1 ˆ e t − 1 + · · · + ρ s ˆ e t − s + v t (18) Perform an LM = T × R 2 test for autocorrelation with the test statistic from the χ 2 ( s ) distribution. (Rejection = autocorrelation)

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Model Selection Choosing the order, p (Lags of Y ) 1 Check correlogram of errors after estimating an AR(1) model, include the significant lags as new Y variables. Repeat with new correlogram. 2 Compute a PACF correlogram of the Y variables and select significant lags Choosing the order, q (Lags of X ) 1 Begin by selecting the maximum length you are willing to consider. 2 Create alternate models of different lag lengths. 3 Using some criterion (AIC, BIC/SC), pick the model with the lowest value

Heteroskedasticity Time Series Endogeneity Simultaneous Equations HAC Standard Errors If you can’t fully eliminate autocorrelated error terms, then you’ll need to compute standard errors that are robust to it. These are called HAC errors. The most commonly used one is the Newey-West Standard Error: ˆ SE NW = ˆ SE HC 1 ( 1 + g ) (19) where ˆ SE HC 1 are the White standard errors from heteroskedasticity, and g is some function of the autocorrelations. In general, putting in HAC standard will lead to wider confidence intervals.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations AR(1) Error to ARDL Model Suppose we write the following model y t = α + β 0 x t + e t (20) but we believe the error terms follow the following process: e t = ρ e t − 1 + v t (21) How would we correctly write the model to get rid of autocorrelation in the errors?

Heteroskedasticity Time Series Endogeneity Simultaneous Equations AR(1) Error to ARDL Model 1 Plug in the error formula into your original equation: y t = α + β 0 x t + ρ e t − 1 + v t 2 Rewrite the error term for t − 1 with the original model e t − 1 = y t − 1 − α − β 0 x t − 1 3 Plug into the model and rearrange terms: y t = α + β 0 x t + ρ ( y t − 1 − α − β 0 x t − 1 ) + v t y t = δ + θ y t − 1 + β 0 x t + β 1 x t − 1 + v t This is now an ARDL(1,1) model where the parameters have some relation to each other.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations ARDL(1,1) to IDL Suppose we had an ARDL(1,1) Model: y t = δ + θ 1 y t − 1 + δ 0 x t + δ 1 x t − 1 + e t (22) and wanted to translate it to an Inifnite Distributed Lag (IDL) model of the following form: y t = α + β 0 x t + β 1 x t − 1 + β 2 x t − 2 + · · · + v t (23) How would we do it?

Heteroskedasticity Time Series Endogeneity Simultaneous Equations ARDL(1,1) to IDL 1 Write the IDL model for y t − 1 : y t − 1 = α + β 0 x t − 1 + β 1 x t − 2 + β 2 x t − 3 + · · · + v t − 1 2 Plug it into your original ARDL model: y t = δ + θ 1 ( α + β 0 x t − 1 + β 1 x t − 2 + β 2 x t − 3 + · · · + v t − 1 ) + δ 0 x t + δ 1 x t − 1 + e t 3 Match up terms according to the lags of the x variables: y t = ( δ + θ 1 α | {z } α ) + δ 0 |{z} β 0 x t + ( δ 1 + θ 1 β 0 | {z } β 1 ) x t − 1 + θ 1 β 1 | {z } β 2 x t − 2 + · · · + u t This process generalizes to different orders of the ARDL model.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Total Effects in IDL Models If we are interested in knowing what a one unit change in x has on future outcomes of y , we would compute the sum: TE = ∞ X k = 0 β k Assuming β k = β 0 r k where | r | < 1, this infinite sum is given by the formula: TE = ∞ X k = 0 β 0 r k = β 0 1 − r This is known as a Geometric Series!

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Forecasting Suppose we have an AR(1) model and we wanted to make a forecast, or prediction, for the next three periods. Assuming our model is correct, what would the next three periods true outcomes be? y t + 1 = θ 0 + θ 1 y t + e t + 1 (24) y t + 2 = θ 0 + θ 1 y t + 1 + e t + 2 (25) y t + 3 = θ 0 + θ 1 y t + 2 + e t + 3 (26) But what would our forecasts be?

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Forecasting In general, our forecasts will be of the form: ˆ y t + s = E [ y t + s |I t ] (27) where I t is all the data you have today. In our example, that creates forecasts ˆ y t + 1 = ˆ θ 0 + ˆ θ 1 y t (28) ˆ y t + 2 = ˆ θ 0 + ˆ θ 1 ˆ y t + 1 (29) ˆ y t + 3 = ˆ θ 0 + ˆ θ 1 ˆ y t + 2 (30) While creating point forecasts might be straightforward, how would we compute the variance of our forecast errors f s = y t + s − ˆ y t + s ?

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Forecast Errors Given the definition of forecast errors, we can begin computing them for our current model: f 1 = y t + s − ˆ y t + s = ( θ 0 − ˆ θ 0 ) + ( θ 1 − ˆ θ 1 ) y t + e t + 1 One simplifying assumption that we use is ˆ θ = θ . This assumption is reasonable as in large samples we have ˆ θ ≈ θ (unbiased), and is useful as it gets rid of a lot of terms. The forecast errors can then written as: f 1 = e t + 1 f 2 = θ 1 ( y t + 1 − ˆ y t + ) + e t + 2 = θ 1 f 1 + e t + 2 f 3 = θ 1 f 2 + e t + 3

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Forecast Errors Forecast errors can then be derived as the following: Var( f 1 ) = Var( e t + 1 ) = σ 2 Var( f 2 ) = Var( θ 1 f 1 + e t + 2 ) = ( θ 2 1 + 1 ) σ 2 Var( f 3 ) = Var( θ 1 f 2 + e t + 3 ) = ( θ 4 1 + θ 2 1 + 1 ) σ 2 We can then construct confidence intervals for our forecasts with: CI s = ˆ y t + 1 ± t c · p Var( f s ) (31)

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Granger Causality Suppose we had an ARDL(p,q) model y t = δ + p X s = 1 θ s y t − s + q X k = 0 δ k x t − k + e t (32) and we wanted to know if the lags of X were impact the outcome of Y . A Granger Causality Test is a F test that tests the joint signiciance of the δ k terms: H 0 : δ 0 = δ 1 = · · · = δ q = 0 H 1 : At least one is not equal to 0

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Exogeneity Given data ( x i , y i ) , we may estimate a linear model: y i = β 1 + β 2 x i + e i (33) Originally, we assumed that the error was uncorrelated with our x variable, i.e our x variable is exogenous : E [ Xe ] = 0 or E [ e | X ] = 0 for all X While this works for many models, this assumption is not always satisfied.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations When Can Endogeneity Fail? 1 Suppose we wanted to estimate the effect of an extra year of education on wages? WAGE i = β 0 + β 1 EDUC i + e i (34) Unobserved "ability" might increase education and earnings! More accomplished students are accepted to universities. More productive people earn a higher wage. 2 Suppose we wanted to estimate the effect of price on the demand of bottled water? Q D i = β 0 + β 1 PRICE i + e i (35) A natural disaster might hit both! If tap water is shut off, people will demand more bottle water. If supply chains are disrupted, then firms might charge more.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Why does it matter? In these cases, we have Cov( X , e ) ̸ = 0, but what affect does this have on our OLS estimates? E [ ˆ b 1 ] = Cov( X , Y ) Var( X ) = Cov( X , β 0 + β 1 X + e ) Var( X ) = Cov( X , β 0 ) Var( X ) + Cov( X , β 1 X ) Var( X ) + Cov( X , e ) Var( X ) = β 1 + Cov( X , e ) Var( X ) When X is endogenous, then our estimator is biased!

Heteroskedasticity Time Series Endogeneity Simultaneous Equations How do we solve it? One solution to the problem of endogeneity are instruments, denoted as Z . They must satisfy the following conditions (Exclusion) Z is uncorrelated with the regression error e , i.e. Cov( Z , e ) = 0 (Relevance) Z is (strongly) correlated with X

Heteroskedasticity Time Series Endogeneity Simultaneous Equations How do we use an IV? Suppose the model is given Y i = β 0 + β 1 X i 1 + · · · + β B X iB + θ 1 W i 1 + · · · + W iG + e i (36) where X is the endogenous variable and W is an exogeneous variables Suppose we have a good instrument for X . How do we use it?

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Solution: 2SLS One solution is the Two Stage Least Squares (2SLS): 1 Estimate the regression of X on Z and W : X i = θ 0 + β 1 Z i + β 2 W i + v i (37) Store the fitted values ˆ X This results in Cov( ˆ X , e ) = 0 2 Estimate the regression of Y on ˆ X and W : Y i = β 0 + β 1 ˆ X i + β 2 W i + e i (38) Your estimates of the second stage will be consistent for β 1 !

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Solution: 2SLS If there’s no additional exogenous variables ( W ), the estimate for β 1 is ˜ b 1 = ∑ N i = 1 ( z i − ¯ z )( y i − ¯ y ) ∑ N i = 1 ( z i − ¯ z )( x i − ¯ x ) (39) The standard error of ˜ b 1 given by Var( ˜ b 1 ) = ∑ ( z i − ¯ z ) 2 ˆ σ 2 IV [ ∑ ( z i − ¯ z )( x i − ¯ x )] 2 ≈ Var( ˆ b OLS 1 ) ( Corr ( X , Z )) 2 (40)

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Solution: 2SLS Some things to keep in mind: 1 You always have to include the exogenous variable W in the first stage. 2 Manually running both models and regressing the second stage on ˆ X is wrong. −→ Parameters are still correct (unbiased/consistent), −→ but standard errors are wrong (inefficient). 3 Want to use a command like lm <- ivreg ( Y ∼ X + W | Z + W , data = . . . )

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Instrument Selection Earlier, we said that Var( ˜ b IV 1 ) ≈ Var( ˆ b OLS 1 ) ( Corr ( X , Z )) 2 so we want to select instruments that are highly correlated with X . This keeps our standard errors as small as possible! With only 1 instrument, a first-stage F -stat greater than 10 is a good rule of thumb With more instruments, you want an even higher F -stat, but no rule of thumb exists.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Checking for Endogeneity (Hausman Test) To check whether X and e are correlated, conduct a Hausman Test 1 Estimate your first stage regression and store the residuals: ˆ v i = X i − ( ˆ θ 0 + ˆ θ 1 Z i + ˆ θ 2 W i ) 2 Include the residuals in the main regression: Y i = β 0 + β 1 X i + δ ˆ v i + e i 3 Conduct the following test: H 0 : δ = 0 (No endogeneity) H 1 : δ ̸ = 0 (Endogeneity)

Heteroskedasticity Time Series Endogeneity Simultaneous Equations How many IVs do we need? Denote L as the amount of instruments we have: L < B : The model is underidentified and the model can’t be estimated. L = B : The model is just-identified and the parameter estimates are unique L > B : The model is overidentified , and problems may arise as your first stage estimate ˆ X approach the original X values in the sample (generally ok).

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Overidentification (Sargan) Tests Only when L > B , you can conduct a Sargan test. This tells you whether you additional instruments should be included. 1 Conduct the 2SLS estimation, and store the residuals of the second stage: ˆ e = ˆ Y − ( ˆ β 0 + ˆ β 1 X + ˆ β 2 W ) (41) 2 Regress ˆ e on all available instruments: ˆ e = θ 0 + θ 1 Z 1 + · · · + θ L Z L (42) 3 If the instruments are all valid, then N × R 2 ∼ χ 2 ( L − B ) 4 If the test statistic is larger than the critical value, then at least one surplus moment condition is not valid.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Simultaneous Equations Simultaneous Equations are used in cases where we believe equilibria occur. Models of supply and demand are a popular case. In general, these models describe systems where multiple endogenous variables are determined simultaneously by a set of equations. Big Topics: 1 Identification 2 Reduced Form 3 Estimation by 2SLS

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Identification Whenever we have a simultaneous equation system, we always break down the variables into endogenous variables and exogenous variables: M Endogenous variables and equations K Exogenous variables For identification (i.e. good estimation), we need each of the M equations to exclude M − 1 exogenous variables.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Reduced Form Equations One way to describe simultaneous equation systems is by solving for the reduced form equations. The reduced form system yields a system of M equations, that each contain one endogenous variable on the left-hand, and all the exogenous variables on the right hand side. When solving for these, treat it as a system of equations where the endogenous variables are the "unknown" variables that you are solving for.

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Reduced Form Equations Structural Equations: y i 1 = α 1 + α 2 y i 2 + α 3 x i 1 + e i 1 y i 2 = β 1 + β 2 y i 1 + β 3 x i 2 + e i 2

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Reduced Form Equations Structural Equations: y i 1 = α 1 + α 2 y i 2 + α 3 x i 1 + e i 1 y i 2 = β 1 + β 2 y i 1 + β 3 x i 2 + e i 2 Reduced Form Equations: y i 1 = π 11 + π 12 x i 1 + π 13 x i 2 + v i 1 y i 2 = π 21 + π 22 x i 1 + π 23 x i 2 + v i 2

Heteroskedasticity Time Series Endogeneity Simultaneous Equations Estimation by 2SLS Estimation of a simultaneous equation system relies on the reduced form equation: 1 Estimate the reduced form equation for the endogenous variable(s) on the right hand side of the equation. Treat these as the first stage. 2 Run a regression of the original equation, replacing the true values of the RHS endogenous variables with their reduced form fitted values If you don’t do this, then estimates will be biased and inconsistent!