Empirical Methods in Finance
A note on asymptotic standard errors
Lecturer: . Lochstoer UCLA Anderson School of Management
December 27, 2021
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1 Departing from the OLS assumptions: Asymptotic Theory
Standard OLS makes strong assumptions. In particular, we assume the residuals are normally distributed and i.i.d. This is not a reasonable assumption for Önancial data. Can we make less assumptions and still maintain the conclusions and intuition from the strict OLS tests? The answer is yes, if we are willing to accept statistics based on asymptotic theory as a good approximation to Önite sample statistics.
The model:
yt=x0t +”t (1)
where we still need the identifying OLS assumption: E [x0t”t] = 0. Let xt and be k 1. We will soon make some assumptions regarding the second moments, but we do not make any distributional
assumptions. We want to Önd the asymptotic distribution of the OLS estimate of k 1 OLS moment conditions for an arbitrary choice of b
ft (b) = xt yt x0tb DeÖne the sample mean of these moment conditions as
gT (b) = T ft (b)
; ^b. DeÖne the (2)
The least squares minimization Önds ^b such that gT ^b = 0.
Sample means are key in asymptotic analysis because we can take advantage of two huge results:
The Law of Large Numbers and the Central Limit Theorem. Therefore we need to look more closely at the properties of the sample mean.
1.1 The Sample Mean
Consider a random n 1 vector y. The sample mean is 1 XT
yT = T yt (4) t=1
In our tests we use sample means and we need their test statistics in order to perform hypothesis
Assume that the variable y in fact has a constant unconditional mean, i.e., E[yt] = (which is an n 1 vector). In this case, the sample mean is an unbiased estimate of the true mean:
1 PT 1 PT E[yT]=E T yt =T E[yt]=.
testing and in order to say anything about the e¢ ciency of an estimate and the power of the tests. 2
Therefore, we need the variance of the sample mean estimate. In order for the coming statistics to be nicely behaved, we assume that y follows a stationary process. In fact, weíll go one step further and assume that yt is covariance-stationary:
E(yt )(yt j )0= j 8j (5) nx
In words, the unconditional covariance matrix of observations j periods apart is only a function of the distance j and not of time t: In this case the variance of the sample mean is given by
2 1 XT ! 1 XT Var(yT) = E4X yt
! 0 3 yt 5
T t=1 T t=1 = 1 T T jjjj
T j= T T (Work out the last equality yourself!).
1.2 Some deÖnitions
Asymptotic analysis is about what happens when the number of observations goes to inÖnity: T ! 1. In particular, what will happen to the mean and variance of the sample mean estimate as T ! 1? We would like the estimate of the sample mean to be consistent, which means that if we had inÖnite amounts of data the estimated sample mean would equal the true sample mean with probability one.
Convergence in Probability: Consider a sequence of random variables xT and an arbitrarily small, positive “. Then if
lim Pr(jxT cj>”)=0 (7) T!1
we say that xT converges in probability to c: plim xT = c, or xT !p c. We say that ^b is a consistent estimator of if ^b !p .
Law of Large Numbers There are several of these, but hereís one that we will use: If xT has mean T and variance 2T and the ordinary limits of T and 2T are c and 0, respectively, xT is a consistent estimator of c.
So, is the sample mean a consistent estimator of the true mean? We need to check the limits of both its expected value and variance. We already found that the expected value was for any
T . The limit of the variance of the sample mean is
lim E (yT )(yT ) = ( 0+2 1+2 2+2 3+:::)
The last equality deÖnes the matrix S as the inÖnite sum of autocovariance matrices. This is an important concept. It is sometimes referred to as the spectral density matrix. For the variance to go to zero as required, S must be Önite. Ergodicity ensures this by requiring that as the distance between two observations gets very large, the covariance between the two goes to zero. In the scalar case, it is su¢ cient that
j <1 (9) j=0
Another important implication of the Law of Large Numbers is that the sample estimate of the covariance matrix is a consistent estimate of the true covariance matrix:
^=1X1 (y y)(y y)0!p
j T t j T t T j
We only need fairly weak restrictions on the process of yt for this to hold (e.g., Önite fourth moments). Essentially, this estimator is a sample mean of a function fT (yt). In fact, any function which satisÖes the law of large numbers has the property that the sample mean is a consistent estimator of the true mean. This is a key property that underlies the Generalized Method of Moments.
Convergence in Distribution Let FxT (x) be the cumulative density function for the ran- dom vector xT . Then if
lim FxT (x) = Fx (x) (11) T!1
xT converges in distribution to x; xT !d x. If the distribution is well known, like the normal distribution, we usually write xT !d N ; 2. Now, weíre ready for the punchline!
pT(yT )!d N(0;S) (12) The (scaled) sample mean estimate converges in distribution to a normal variable with mean zero
and variance equal to the inÖnite sum of autocovariances. This is what makes asymptotic analysis 4
Central Limit Theorem
beautiful!
We can extend this result to any function of yT using the Delta Method: Let c (yT ) be a j 1
continuous and di§erentiable function of the k 1 vector yT . Then
pT(c(yT) c())!d N0;C()SC()0 (13)
where C () plim @c jy =. For brief intuition on this result, consider a Taylor-expansion of @yT T
c (). As T gets large, the Örst order term dominates. An a¢ ne function of a normally distributed variable (yT , in this case) is also normal. The Delta method is very useful. If we can estimate parameters based on the sample mean of moments that perhaps are nonlinear in the parameters, the Delta method gives us the limiting distribution of the parameters. We use the distribution of the estimated parameters for hypothesis testing.
Asymptotic Distribution We use asymptotic theory by assuming that the asymptotic dis- tribution is a good proxy for the true Önite sample distribution. If
pT^ 0!d N(0;V) (14) Then the asymptotic distribution of ^ is
^!a N;T1V (15) The asymptotic covariance matrix is then T1 V , which must be estimated for instance by the sample
covariance matrix.
1.3 Asymptotic OLS continued
Now, letís apply this to the OLS example. Remember, the model is
yt=x0t +"t (16)
where E [xt"t] = 0, and xt is k 1. We can rewrite this model as
E[ft( )]=0; ft(b)xt yt x0tb (17)
DeÖne sample mean of ft evaluated at an arbitrary b as 1 XT
gT (b) T ft(b) (18) t=1
Minimizing the sum of squared errors is equivalent to Önding the ^b that sets the sample mean to
We want the distributional properties of the estimator so we can perform hypothesis tests. Apply the Delta method by applying a Taylor expansion to g :
gT ^bT=gT ( )+dT ( )^bT +higher order terms (21) where dT ( ) @gT (b) jb= . Drop the higher order terms and solve for ^b :
^bT = [dT()] 1gT() (22) Now, weíre ready for the Delta method, which says that
gT ^b = 0 (19) We know that the true, unobserved sample mean converges to zero:
pT(gT ( ) 0)!d N(0;S); whereS= X1 Eft( )ft j( )0 (20) j = 1
pT^bT !d N0;d( ) 1Sd( ) 1 (23) andS= P1 Eft()ft j()0.FromthedeÖnitionofft(b)we
which gives
dT (b) = xtx0t, so plim dT (b) = E xtx0t (24) t=1
pT ^bT !d N 0; E xtx0t 1 SE xtx0t 1 (25)
^bT !aN 0;T1Extx0t 1SExtx0t 1 (26)
1.4 Estimating the Variance-covariance matrix
What is S in the OLS case?
S = X1 E f t ( ) f t j ( ) 0
= 1 E xtx0t j"t"t j (27)
To put this into use, we need an estimate of the asymptotic covariance matrix. Here are three
commonly used:
I.i.d. residuals with constant variance 2" This strong assumption allows us to drop all
terms where j 6= 0.
Siid =Extx0t2 (28) Asy:V ar ^b = T1 E xtx0t 1 2 (29)
A consistent estimator of this matrix is
Est:Asy:V ar ^b = T1 XT0 XT 1 ^"0T ^"T (30) Independent residuals, but squared value correlated with xís (White (1980)) This
case allow for time-varying volatility (heteroskedasticity) of the residuals
S = E xtx0t"2t (31)
Asy:V ar ^b = T1 E xtx0t 1 E xtx0t"2t E xtx0t 1 (32) A consistent estimator of this matrix is
^ 1 XT Est:Asy:V ar b = XT0 XT xtx0t^"2t
Non-zero correlations between ft (b) and ft j (b) if jjj q.
^ 1 0 1 0@ Xj 0 1A 0 1
Asy:V ar b = T E xtxt E xtxt j"t"t j E xtxt (34) q= j
where the estimate of this is the sample analogue. This estimate is due to Hansen and Hodrick (1983). However, in Önite samples the estimate of the variance-covariance matrix is not always positive-deÖnite (i.e., we sometimes get negative variance). Newey-West (1987) has proposed a much used correction that alleviates this problem:
v=1 q+1 The estimate covariance matrix is then
ft (b)ft v (b)0 (35) S^T =RT 0;^bT+Xq q+1 vRT v;^bT+RT v;^bT0 (36)
Est:Asy:V ar ^b = T XT0 XT 1 S^T XT0 XT 1 (37) Correcting for autocorrelation is especially important when using overlapping observations (e.g.,
annual overlapping at the monthly frequency).
1.5 Standard OLS vs. robust OLS
Note that in this case, the empirical implementation of the asymptotic (robust) OLS is the same as the Önite sample OLS, but without having to make the normality assumption! Since the robust OLS estimates all are normal, the joint test statistics are distributed 2: This is the same as the Önite sample distribution with known covariance matrix. Of course, we never have inÖnite amounts of data as we implicitly assume when we use asymptotic theory in practice. Since we do not, we cannot know the covariance matrix. We know, however, that we need quite a few datapoints to be able to estimate a covariance matrix with reasonable accuracy.
RT (v;b) = T where the estimate of the spectral density matrix is
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