Chapter 3 Linear Econometrics for Finance
Robust asymptotic inference
We considered a general linear regression model.
We derived consistency and asymptotic normality of the parameter estimates. We tested hypothesis.
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Issue: our tests (and their outcomes) would be different if Assumption 2(a) (namely, E(εt|xt, xt−1, …εt−1, εt−2, …) = 0 – uncorrelatedness of the error terms) and Assump- tion2(b)(namely,E(ε2t|xt,xt−1,…εt−1,εt−2,…)=σε2 -homoskedasticity)werenot satisfied.
Why? The asymptotic variance of the β estimator would be different and all our tests rely on it!
Important: both Assumption 2(a) and Assumption 2(b) might be severely violated in finance. Now, we discuss a way to handle their violation.
Autocorrelation and Heteroskedasticity robust stan- dard errors (HAC standard errors)
to our problem:
T −1 T √1′√1
T β−β = T xtxt T T xtεt
T −1 T
1′ 1 =Txtxt √Txtεt
t=1 t=1 T
d ′−11′−1
Y N 0,E(xtxt) Var √T xtεt E(xtxt) .
Chapter 3 Before:
Linear Econometrics for Finance
Var √ 1 T
= T1 E(xtx′tε2t ) − E(xtεt)E(xtεt)′ (2)
= σε2 E (xt x′t ), (3)
= T Var(xtεt) t=1
• Eq. (1) derives from Assumption 1 and Assumption 2(a). Notice, in fact, that Cov(xtεt,x′t+jεt+j) = E(xtεtx′t+jεt+j) − E(xtεt)E(x′t+jεt+j) = E(xtεtx′t+jεt+j) be- cause E(xtεt) = E(xtE(εt|xt)) = 0 given Assumption 1. In addition, E(xtεtx′t+jεt+j) = E(xtεtx′t+jE(εt+j|xt+j,xt,εt)) = 0 for j > 0 given Assumption 2(a). As mentioned, Assumption 2(a) is an assumption of uncorrelatedness of the error terms.
• Eq. (2) is the matrix analogue of the “alternative variance formula” (V ar(x) = E(x2) − (E(x))2)
• Eq. (3) derives from E(xtεt) = E(xtE(εt|xt)) = 0 (Assumption 1) and E(xtx′tE(ε2t |xt)) = σε2E(xtx′t), which is the result of the homoskedasticity condition in Assumption 2(b).
√ T β−β
Yd N(0, E(xtx′t)−1σε2E(xtx′t)E(xtx′t)−1) = N(0, E(xtx′t)−1σε2E(xtx′t)E(xtx′t)−1)
= N 0, σε2E(xtx′t)−1 . Now, let us relax the assumptions and go step-by-step.
Linear Econometrics for Finance
The logic, first. Assume at is a stationary time-series and is such that E(at) = 0. We will return to the concept of “stationarity” in the next chapter. For now, we will just say that stationarity implies that the first and second moment do not change over time. Consider
t=1 Var√2 = 2E at
= 12E(a21 + a2 + a1a2 + a2a1)
Now, consider
= 21 2E(a2t ) + (E(atat+1) + E(atat−1)) . at 132
Var√3 = 3E at t=1
More generally,
= 31E(a1 +a2 +a3)(a1 +a2 +a3)
= 31E(a21 +a2 +a23 +a1a2 +a1a3 +a2a1 +a2a3 +a3a1 +a3a2)
= 31(3E(a2t ) + 2 (E(atat+1) + E(atat−1)) + E(atat+2) + E(atat−2)).
Chapter 3 Linear Econometrics for Finance
Var √k = k[(kE(at)+(k−1)(E(atat+1)+E(atat−1))
Naturally, as k → ∞, then k
+(k − 2)(E(atat+2) + E(atat−2) +… + E(a1ak) + E(aka1)]
k k − | j | E ( a t a t + j ) . j=−k k
Var √k =
E(atat+j) Y E(atat+j). j=−∞
Back to our problem. In the more general case in which we drop Assumption 2(a) and Assumption 2(b), we obtain (by setting xtεt = at):
T −1 T √1′1
T β−β = T xtxt √T xtεt t=1 t=1
d ′−1 1 ′−1
YN(0,E(xtxt) Var √T xtεt E(xtxt) ) t=1
∞ = N 0, E(xtx′t)−1 E(εtxtx′t+j εt+j )E(xtx′t)−1 .
This is the expression for the variance-covariance matrix of the OLS estimates in the case
of correlated and heteroskedastic ε’s.
Note. Accounting for autocorrelation and heteroskedasticity generally increases the stan-
Chapter 3 Linear Econometrics for Finance dard errors. If you do not correct the standard errors you might obtain over-rejections of
the null of no dependence, i.e., β = 0 – worth thinking about … 1.1 Sub-cases
• It is now trivial to see what happens when we strengthen the assumptions. 1. Uncorrelated errors: E(εt|xt, xt−1, …εt−1, εt−2, …) = 0. Then,
E(εtxtx′t+jεt+j) = j =−∞
E(εtxtx′tεt) + E(εtxtx′t+1εt+1) + E(εtxtx′t−1εt−1)
E(εtxtx′tεt) + E(εtxtx′t+1E(εt+1|xt+1, εt, …)) +E(E(εt|xt, εt−1, …)xtx′t−1εt−1)
E(xtx′tε2t ).
If the errors are just uncorrelated, but heteroskedastic, we obtain
√ d ′−1 ′2 ′−1 T β − β Y N 0, E(xtxt) E(xtxtεt )E(xtxt) .
2. Uncorrelated and homoskedastic errors: E(εt|xt, xt−1, …εt−1, εt−2, …) = 0 and E(ε2t |xt, xt−1, …εt−1, εt−2, …) = σε2. Then,
E(εtxtx′t+jεt+j) j =−∞
= E(xtx′tε2t)
= E(xtx′tE(ε2t |xt, xt−1, …εt−1, εt−2, …))
= E(xtx′tσε2) = σε2E(xtx′t).
Chapter 3 Linear Econometrics for Finance Again, under these assumptions,
√ T β−β
Yd N 0, E(xtx′t)−1σε2E(xtx′t)E(xtx′t)−1 Yd N 0, σε2E(xtx′t)−1 .
Estimation of the asymptotic variance-covariance matrix
The general case:
V ar(β) = 1 E(xtx′t)−1 E(εtxtx′t+j εt+j )E(xtx′t)−1.
T j=−∞ Why do we divide by T?
How do we estimate ∞j =−∞ E (εt xt x′t+j εt+j ) with data? This is an infinite sum. Problematic, right?
However, this sum is nothing but the asymptotic (for k → ∞) limit of k k−|j| E(εtxtx′t+j εt+j ).
Easy, just replace the expectation with sample analogues!
T Write k k−|j| 1 (εx x′ ε )
and, in principle, let k,T → ∞ at some
rate! Hence,
t t t+j t+j t=1
V ar(β) = 1 1 (x x′ )
Linear Econometrics for Finance
k T ttt+jt+j T tt j=−k t=1 t=1
1 (ε x x′ ε ) 1 (x x′ )
• Tests would now be constructed using V ar(β).
• One can show that V ar(β) →p V ar(β) (i.e., V ar(β) is ”consistent” for V ar(β)) as k,T→∞withk →0andk2 →∞.
• This (heteroskedasticity and autocorrelation consistent) variance estimator is gen- erally called HAC (Hetero… and AutoCorrelation consistent) variance estimator or Newey-West variance estimator after Newey and West (1987).
• This is what softwares (like e-views) compute (to find standard errors) when you select this option!
• A short-cut (Hansen and Hodrick, 1980):
1 (ε x x′ ε
T t t t+j t+j
2. The heteroskedastic (but uncorrelated) case:
V ar(β) = 1 1 (x x′ )
1 (x x′ )
• Estimate by
T −1 T T 11′12′1′
(x x ) (ε x x ) (x x ) T Ttt
t=1 t=1 t=1
V ar(β) = 1 E(xtx′t)−1E(xtx′tε2t )E(xtx′t)−1. T
• This estimated variance is called “heteroskedasticity consistent” or White variance estimator after White (1980).
Chapter 3 Linear Econometrics for Finance
• This is, again, what softwares (like e-views) compute (to find standard errors) when you select this option!
For some discussion, see Cochrane (Chapter 11).
Chapter 3 Linear Econometrics for Finance
3 Regression, predictability and robust standard er- rors
We will see later that it is standard to predict stock returns using financial ratios, like the dividend-to-price ratio or the book-to-market ratio. We will also discuss the economic justification underlying this approach. The file “predictability.xls” contains monthly data on returns on the value-weighted market portfolio, returns on the equally-weighted mar- ket portfolio, the t-bill rate, the dividend-to-price ratio (dyny), the earnings-to-price ratio (epny) and the book-to-market ratio (bmny).
We will run a regression of the excess (with respect to the risk-free rate) continuously- compounded returns on the value-weighted market portfolio on the past dividend-to-price ratio and see what we obtain. The regression is:
M f d log(1+Rt )−log(1+Rt)=β0 +β1 × p
Chapter 3 Linear Econometrics for Finance
Dependent Variable: FIRST
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample (adjusted): 1946M02 2001M12
Included observations: 671 after adjustments
FIRST = C(1)+C(2)*DYNY(-1)
Coefficient C(1) −0.006953
C(2) 0.326995
Std. Error t-Statistic 0.005130 −1.355507
0.129587 2.523365 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Prob. 0.1757
0.0119 0.005361
0.041117 −3.549815 −3.536377 −3.544610
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.009428 0.007947 0.040953 1.122003 1192.963 6.367373 0.011854
spite of a low R2. returns, let us predict yearly excess returns. In other word:
log(1+RM )−log(1+Rf ) =β +β ×
Notice the significance of the predictor in
Let us now consider the same regression, but rather than predicting monthly excess
t+j t+j 0 1
Chapter 3 Linear Econometrics for Finance
Dependent Variable: SECOND
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample: 1946M12 2001M12
Included observations: 661
SECOND = C(1) + C(2)*DYNY(-11)
Coefficient C(1) −0.008022
C(2) 3.452912
Std. Error t-Statistic 0.017471 −0.459158
0.438622 7.872173 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Prob. 0.6463
0.0000 0.123122
0.141469 −1.158808 −1.145211 −1.153538
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.085955 0.084568 0.135355 12.07355 384.9860 61.97110 0.000000
The significance of the slope estimate has gone up drastically. The typical conclusion drawn in the literature (and in the industry) is that yearly returns are easier to predict than monthly returns.
What happens if, instead, we compute the standard errors using Newey-West? Let us see.
Chapter 3 Linear Econometrics for Finance
Dependent Variable: SECOND
Method: Least Squares (Gauss-Newton / Marquardt steps)
Date: Time:
Sample: 1946M12 2001M12
Included observations: 661
HAC standard errors & covariance (Bartlett kernel, Newey-West fixed bandwidth = 7.0000)
SECOND = C(1) + C(2)*DYNY(-11)
Coefficient C(1) −0.008022
C(2) 3.452912
Std. Error t-Statistic 0.039249 −0.204391
0.946617 3.647634 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat -statistic
Prob. 0.8381
0.0003 0.123122
0.141469 −1.158808 −1.145211 −1.153538
0.174784 13.30524
Adjusted R-squared S.E. of regression
Sum squared resid
Log likelihood F-statistic Prob(F-statistic) Prob( -statistic)
0.085955 0.084568 0.135355 12.07355 384.9860 61.97110 0.000000 0.000286
The use of Newey-West standard errors has drastically reduced the statistical signifi- cance of the slope estimates. Yet, significance is – again – higher than for the monthly regressions. Again, predictability is stronger over the long haul. Let us now go to 5 years.
Chapter 3 Linear Econometrics for Finance
Dependent Variable: THIRD
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample: 1950M12 2001M12
Included observations: 613
THIRD = C(1)+C(2)*DYNY(-59)
Coefficient C(1) 0.097093
C(2) 13.86935
Std. Error 0.033217
t-Statistic 2.923009
Prob. 0.0036
0.0000 0.647196
0.262306 −0.227127 −0.212711 −0.221520
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.325248 0.324144 0.215643 28.41269 71.61434 294.5180 0.000000
0.808166 17.16153 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Statistical significance has increased further. What about if we use Newey-West standard errors now? Again, we will see some attenuation which is not going to change the overall picture.
Chapter 3 Linear Econometrics for Finance
Dependent Variable: THIRD
Method: Least Squares (Gauss-Newton / Marquardt steps)
Date: Time:
Sample: 1950M12 2001M12
Included observations: 613
HAC standard errors & covariance (Bartlett kernel, Newey-West fixed bandwidth = 6.0000)
THIRD = C(1)+C(2)*DYNY(-59)
Coefficient C(1) 0.097093
C(2) 13.86935
Std. Error t-Statistic 0.093247 1.041236
1.972175 7.032515 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat -statistic
Prob. 0.2982
0.0000 0.647196
0.262306 −0.227127 −0.212711 −0.221520
0.087148 49.45627
Adjusted R-squared S.E. of regression
Sum squared resid
Log likelihood F-statistic Prob(F-statistic) Prob( -statistic)
0.325248 0.324144 0.215643 28.41269 71.61434 294.5180 0.000000 0.000000
How do we explain all of these results?
3.1 A traditional predictive system for stock market returns
• Consider the following predictive system:
yt+1 = α+βxt+ut+1
xt+1 = ρxt+εt+1,
where yt is an excess stock return and xt is a de-meaned predictor (for example,
like before, the dividend-to-price ratio). 14
Chapter 3 Linear Econometrics for Finance
• Potentially small estimated slopes (and low R2 values) at high frequencies translate (mathematically) into high estimated slopes (and high R2 values) at low frequencies if ρ is high and the correlation between ut+1 and εt+1 is negative, as in the data. In fact, write:
yt+1 +yt+2
= α+βxt +ut+1 +α+βxt+1 +ut+2
= 2α+β(xt +xt+1)+ut+1 +ut+2
= 2α+β(1+ρ)xt +βεt+1 +ut+1 +ut+2.
• More generally, for an horizon of forward aggregation h, we have β = β(1 + ρ + …. + ρh−1).
• What do we learn? Aggregation increases the true (and estimated) slope more than it increases the size of the slope estimator’s standard error. Hence, long-run predictability is more statistically significant than short-run predictability. For a recent discussion, see Bandi, Perron, Tamoni and Tebaldi (2019).
• Since the error terms from the long-run regressions have a “moving average” struc- ture (i.e., βεt+1 + ut+1 + ut+2, if we only aggregate over two periods), the use of Newey-West standard errors is important. (The precise meaning of the terminology “moving average” will be discussed in the next chapter.)
• Notice that the dividend-to-price ratio is modeled as an autoregressive process. We turn to time-series modeling (and autoregressive dynamics, among other specifica- tions) in what follows.
Chapter 3 Linear Econometrics for Finance
References
[1] Bandi, F.M., Perron, B., Tamoni, A. and C. Tebaldi (2019). The scale of predictability. Journal of Econometrics 208, 120-140.
[2] Cochrane, J. (2001). Asset Pricing. Princeton University Press.
[3] Hansen, L.P. and R.J. Hodrick (1980). Forward exchange rates as optimal predictors of future spot rates: an econometric analysis. Journal of Political Economy 88, 829- 853.
[4] Newey, W. and K.D. West (1987). A simple, positive-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-708.
[5] White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817-838.
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