Lecture 3: Serial Correlation

homoskedasticity assumption

variance of ut given xt is constant, no correlation between errors

Var(ut | xt) = σ2

Cov(ut, us | xt, xs) = 0

serial correlation definition

when error terms are correlated across time

why does serial correlation happen

unobserved factors can influence outcome variable across multiple time periods

eg weather in ice cream consumption; if not included then errors are correlated

consequences of serial correlation

OLS still unbiased (under strict exogeneity)

OLS no longer BLUE (GLS has lower variance)

standard errors and test statistics incorrect

model of serial correlation (ie, error term is AR(1))

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effect of serial correlation on variance of B hat

if p=0 (no serial correlation) the variance of OLS is the standard OLS variance

but variance estimator is biased when p=/=0

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serial correlation in the presence of lagged dependent variables (how it can have zero conditional mean error process but still have serial corr in errors)

error term not systematically related to yt-1

even though ut not correlated with yt-1, can still be correlated with yt-2 or more

→ show that this means that ut and ut-1 can still be correlated

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show omitted variable yt-2 in model and how to fix for this

include it in the regression

sub in the ut to regression

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why do we test for serial correlation

bias in standard errors = unreliable tests

OLS is no longer BLUE

testing for serial correlation assumptions (strong exogeneity) and why do we need them

null: p = 0 (no serial correlation)

strong exogeneity/homoskedasticity assumptions ensure OLS estimates are asymptotically normal - can perform valid hypothesis tests

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construct test for serial correlation w strong exogeneity

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estimate regression model for b hat coefficients

compute estimated residuals

regress estimated residuals on lagged values

t test for p

downsides of test for serial correlation (strong exogeneity)

only detects first order serial correlation

serial correlation violates strong exogeneity (if it exists, test is invalid)

testing for serial correlation without strong exogeneity

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run OLS of yt on xt to get b hat estimates

compute u hat

regress u hat on explanatory variables and lagged residual ut-1 (this controls for correlation between regressors and past errors and makes test valid even if strong exogeneity fails)

test for serial corrrelation (H0 p=0)

testing for higher order serial correlation and why would you do this

why: if errors are correlated over longer lags

obtain b hat

compute ut hat

run auxiliary regression including all explanatory variables and lagged residuals up to q periods

conduct F test for joint significance of p1 p2 … pq

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quasi differencing approach to correct for serial correlation (how to do and disadvantages)

yt - p(yt-1)

loses period t=1 (efficiency loss)

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what to do with variance of first period when quasi differencing

adjust for fact that var(u1) =/= var(et)

multiply by sqrt(1-p²) to standardise variance

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quasi differencing procedure

estimate p from OLS residuals of original model

transform using quasi differencing

run OLS on transformed model

use standard errors corrected for serial correlation to perform valid inference

feasible generalised least squares FGLS why we use

since p is unknown, need to estimate it then apply the quasi differencing transformation

more efficient than OLS but not necessarily BLUE

FGLS procedure

obtain b hat estimates

estimate p using residuals

define quasi differencing using p hat

run OLS on final model

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