chapter 15: estimation of dynamic causal effects · is not the usual one from cross-sectional...

Chapter 15: Estimation of Dynamic Causal Effects

Copyright © 2011 Pearson Addison-Wesley. All rights reserved. 15-11-1

Outline

1. Dynamic Causal Effects and the Orange Juice Data

2. Estimation of Dynamic Causal Effects with Exogenous Regressors: The Distributed Lag Model

3. HAC Standard Errors

Copyright © 2011 Pearson Addison-Wesley. All rights reserved. 15-2

3. HAC Standard Errors

4. Application to Orange Juice Prices

5. More on Exogeneity

Dynamic Causal Effects & Orange Juice Data

A dynamic causal effect is the effect on Y of a change in X over time.

For example:

• The effect of an increase in cigarette taxes on cigarette


• The effect of an increase in cigarette taxes on cigarette consumption this year, next year, in 5 years;

• The effect of a change in the Fed Funds rate on inflation, this month, in 6 months, and 1 year;

• The effect of a freeze in Florida on the price of orange juice concentrate in 1 month, 2 months, 3 months…

The Orange Juice Data

• Monthly, Jan. 1950 – Dec. 2000 (T = 612)

• Price = price of frozen OJ (a sub-component of the producer price index; US Bureau of Labor Statistics)

• %ChgP = percentage change in price at an annual rate, so %ChgP = 1200∆ln(Price )


%ChgPt = 1200∆ln(Pricet)

• FDD = number of freezing degree-days during the month, recorded in Orlando FL

– Example: If November has 2 days with lows < 32o, one at 30o and at 25o, then FDDNov = 2 + 7 = 9

Initial OJ regression

= -.40 + .47FDDt(.22) (.13)

• Statistically significant positive relation

• More freezing degree days � price increase

�%t

ChgP


• Standard errors are heteroskedasticity and autocorrelation-consistent (HAC) SE’s (more on this later)

• But what is the effect of FDD over time?

Dynamic Causal Effects

Example: What is the effect of fertilizer on tomato yield?

An ideal randomized controlled experiment

• Fertilize some plots, not others (random assignment)

• Measure yield over time – over repeated harvests – to estimate causal effect of fertilizer on:


estimate causal effect of fertilizer on:

– Yield in year 1 of experiment

– Yield in year 2, etc.

• The result (in a large experiment) is the causal effect of fertilizer on yield k years later.

Dynamic causal effects, ctd.

In applications, we cannot conduct such ideal randomized controlled experiments. Here:

• We only have one US OJ market (unlike multiple plots)

• We cannot randomly assign FDD to different replicates of the US OJ market (what does this even mean?)


US OJ market (what does this even mean?)

• We cannot measure the average (across “subjects”) outcome at different times – there is only one “subject”!

So we cannot estimate the causal effect at different times using the differences estimator

Dynamic causal effects, ctd.

An alternative thought experiment:

• Randomly give the same subject different treatments (FDDt) at different times

• Measure the outcome variable (%ChgPt)

• The “population” of subjects consists of the same subject (OJ


• The “population” of subjects consists of the same subject (OJ market) but at different dates

• If the “subjects” (the subject at different times) are drawn from the same distribution – that is, if Yt, Xt are stationary– then the dynamic causal effect can be deduced by OLS regression of Yt on lagged values of Xt.

• This estimator (regression of Yt on Xt and lags of Xt) is called the distributed lag estimator.

Dynamic Causal Effects and the Distributed Lag Model

The distributed lag model is:

Yt = β0 + β1Xt + … + βrXt–r + ut

• β1 = impact effect of change in X = effect of change in Xt

on Yt, holding constant past Xt

• β = 1-period dynamic multiplier = effect of change in X


• β2 = 1-period dynamic multiplier = effect of change in Xt–1

on Yt, holding constant Xt, Xt–2, Xt–3,…

• β3 = 2-period dynamic multiplier (etc.) = effect of change in Xt–2 on Yt, holding constant Xt, Xt–1, Xt–3,…

• Cumulative dynamic multipliers– The h-period cumulative dynamic multiplier is β1 + … + βh+1 = impact

effect + … + (h+1)-period effect

– The long-run cumulative dynamic multiplier is β1 + … + βr+1 (note, r = last lag in model)

Exogeneity in Time Series Regression

Exogeneity (past and present)

X is exogenous if E(ut|Xt, Xt–1, Xt–2,…) = 0.

So error term is uncorrelated with regressor in present and past

Strict Exogeneity (past, present, and future)


X is strictly exogenous if E(ut|…, Xt+1, Xt, Xt–1, …) = 0

So error term is uncorrelated with regressor at all times

• Strict exogeneity implies exogeneity

• For now we suppose that X is exogenous – “strict” later

• If X is exogenous, then OLS is appropriate to estimate the dynamic causal effect on Y of a change in X….

Estimation of Dynamic Causal Effects with Exogenous Regressors

Distributed Lag Model:

Yt = β0 + β1Xt + … + βr+1Xt–r + ut

The Distributed Lag Model Assumptions


1. E(ut|Xt, Xt–1, Xt–2,…) = 0 (X is exogenous)

2. (a) Y and X have stationary distributions;(b) (Yt, Xt) and (Yt–j, Xt–j) become independent as j�∞

3. Y and X have eight nonzero finite moments

4. There is no perfect multicollinearity.

The distributed lag model, ctd.

• Assumptions 1 and 4 are familiar

• Assumption 3 is familiar, except for 8 (not four) finite moments – this has to do with HAC estimators

• Assumption 2 is different – before it was (Xi, Yi) are i.i.d. –with time series data, temporal dependence is a reality


2. (a) Y and X have stationary distributions;

• If so, the coefficients do not change within the sample (internal validity);

• and the results can be extrapolated outside the sample (external validity).

• This is the time series counterpart of the “identically distributed” part of i.i.d.

The distributed lag model, ctd.

2. (b) (Yt,Xt) and (Yt–j, Xt–j) become independent as j�∞

– Intuitively, this says that we have separate experiments for time periods that are widely separated.

– For cross-sectional data, we assumed that Y and X were i.i.d., justified by simple random sampling … enabling CLT


i.i.d., justified by simple random sampling … enabling CLT

– A version of the CLT holds for time series variables that become independent as their temporal separation increases –2(b) is the analogue of the “independently distributed” part of i.i.d.

Under the Distributed Lag Model Assumptions:

• OLS yields consistent estimators of β1, β2,…, βr• In large samples, the sampling distribution of is normal

• Yet the formula for the variance of this sampling distribution is not the usual one from cross-sectional (i.i.d.) data, because u is serially correlated, not i.i.d.

iβ̂


because ut is serially correlated, not i.i.d.

• So the usual OLS standard errors are wrong (as in STATA printout under “r” option)

• We need SEs that are robust to autocorrelation and to heteroskedasticity…

Heteroskedasticity and Autocorrelation-Consistent (HAC) Standard Errors

The math, for a single regressor Xt: Yt = β0 + β1Xt + utThe OLS estimator: From SW, App. 4.3,

– β1 =

1

T( X

t− X )u

tt=1

T

∑

1( X − X )

2T

∑


≅ (in large samples)

where vt = (Xt – )ut.

T( X

t− X )

2

t=1

∑

1

Tv

tt=1

T

∑

σX

2

X

HAC standard errors, ctd.

Thus, in large samples,

var( ) = /

= /

var1

Tv

tt=1

T

∑

(σ X

2)

2

1

T2

cov(vt,v

s)

s=1

T

∑t=1

T

∑ (σ X

2)

2


In i.i.d. cross sectional data, cov(vt, vs) = 0 for t ≠ s, so

var( ) = )/ =

This is our usual cross-sectional result (SW App. 4.3).

T s=1t=1

1

T2

var(vt)

t=1

T

∑ (σ X

2)

2

σv

2

T (σx

2)

2


But in time series data, cov(vt, vs) ≠ 0 is typical.

Consider T = 2:

= ¼[var(v1) + var(v2) + 2cov(v1,v2)]

= ½ + ½ρ1 (ρ1 = corr(v1,v2))

var1

Tv

tt=1

T

∑

σ v

2


= ½ ×f2, where f2 = (1+ρ1)

• In i.i.d. data, ρ1 = 0 so f2 = 1, yielding the usual formula

• In time series data, if usual formula off by factor f2

σ v

2

Expression for var( ), general T

Let be the jth-autocorrelation of

var( ) = × fT

where

1

T

σv

2

(σX

2)

2

),(: jttj vvcorr −=ρ ttt uXXv )(: −=


where

fT = (SW, eq. (15.13))

In short, usual formula for SE is corrected by including factor fT !

X

1+ 2T − j

T

ρ

jj=1

T −1

∑

HAC Standard Errors

• Problem: We do not know the correction factor!

• It involves the population’s autocorrelations

• Now,

– In panel data, the factor fT is (implicitly) estimated by using “cluster” – requires n large.

– In time series data, we need a different formula – we


– In time series data, we need a different formula – we must estimate fT explicitly

• Need a consistent estimator of fT• Heteroskedasticity and Autocorrelation Consistent (HAC)

standard errors are those that use consistent estimators of fT


var( ) = ×fT , where fT =

The most commonly used estimator of fT is:

= (Newey-West)

1

T

σv

2

(σX

2)

2

1+ 2T − j

T

ρ

jj=1

T −1

∑

ˆT

f1

1

1 2m

j

j

m j

mρ

−

=

− + ∑ �


where

• This is the “Newey-West” HAC SE estimator

• m is called the truncation parameter

• What m should we choose? Rule of thumb: m = 0.75T1/3

1j m=

ttt

T

t

t

T

jt

jttj uXXvvvv ˆ)(:ˆ with ˆ/ˆˆ:~

1

2

1

−== ∑∑=+=

−ρ

Example: OJ and HAC estimators

. reg dlpoj fdd if tin(1950m1,2000m12), r; NOT HAC SEs

Linear regression Number of obs = 612

F( 1, 610) = 12.12

Prob > F = 0.0005

R-squared = 0.0937

Root MSE = 4.8261

------------------------------------------------------------------------------

| Robust


| Robust

dlpoj | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

fdd | .4662182 .1339293 3.48 0.001 .2031998 .7292367

_cons | -.4022562 .1893712 -2.12 0.034 -.7741549 -.0303575

------------------------------------------------------------------------------

Rerun regression, with Newey-West SE:

newey dlpoj fdd if tin(1950m1,2000m12), lag(7);

Number of obs = 612

maximum lag: 7 F( 1, 610) = 12.23

Prob > F = 0.0005

------------------------------------------------------------------------------

| Newey-West


-------------+----------------------------------------------------------------

fdd | .4662182 .1333142 3.50 0.001 .2044077 .7280288


fdd | .4662182 .1333142 3.50 0.001 .2044077 .7280288

_cons | -.4022562 .2159802 -1.86 0.063 -.8264112 .0218987

------------------------------------------------------------------------------

Uses autocorrelations up to m = 7 to compute the SEs

rule-of-thumb: 0.75*(612)^(1/3) = 6.4 » 7, rounded up a little.

Difference in SEs turned out small – in 4th decimal

. gen l0fdd = fdd; generate lag #0

. gen l1fdd = L1.fdd; generate lag #1

. gen l2fdd = L2.fdd; generate lag #2

. gen l3fdd = L3.fdd; .



. gen l6fdd = L6.fdd;

Example: OJ and HAC estimators in STATA, ctd.

. global lfdd6 "fdd l1fdd l2fdd l3fdd l4fdd l5fdd l6fdd”;

. newey dlpoj $lfdd6 if tin(1950m1,2000m12), lag(7);

Number of obs = 612

maximum lag : 7 F( 7, 604) = 3.56

Prob > F = 0.0009

------------------------------------------------------------------------------

| Newey-West




-------------+----------------------------------------------------------------

fdd | .4693121 .1359686 3.45 0.001 .2022834 .7363407

l1fdd | .1430512 .0837047 1.71 0.088 -.0213364 .3074388

l2fdd | .0564234 .0561724 1.00 0.316 -.0538936 .1667404

l3fdd | .0722595 .0468776 1.54 0.124 -.0198033 .1643223

l4fdd | .0343244 .0295141 1.16 0.245 -.0236383 .0922871

l5fdd | .0468222 .0308791 1.52 0.130 -.0138212 .1074657

l6fdd | .0481115 .0446404 1.08 0.282 -.0395577 .1357807

_cons | -.6505183 .2336986 -2.78 0.006 -1.109479 -.1915578

------------------------------------------------------------------------------

global lfdd6 defines a string which is all the additional lags

Run in this way, output includes coefficients on older lags as well

What are the estimated dynamic multipliers (dynamic effects)? Just add!

FAQ: Do I need to use HAC SEs when I estimate an AR or an ADL model?

A: NO

• The problem to which HAC SEs are the solution only arises when ut is serially correlated: if ut is serially uncorrelated, then OLS SE’s are fine


• In AR and ADL models, (X = past Y), the errors are serially uncorrelated if you have included enough lags of Y

If you include enough lags of Y, then the error term can’t be predicted using past Y, or equivalently by past u – so u is serially uncorrelated

Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors

• X is strictly exogenous if E(ut|…,Xt+1, Xt, Xt–1, …) = 0

• If X is strictly exogenous, there are more efficient ways to estimate dynamic causal effects than by a distributed lag regression:

– Generalized Least Squares (GLS) estimation


– Autoregressive Distributed Lag (ADL) estimation

• But strict exogeneity is rarely plausible in practice

• So we won’t cover GLS or ADL estimation of dynamic causal effects – for details, see SW Section 15.5.

No strict exogeneity in weather/OJ data

• X is strictly exogenous if E(ut|…,Xt+1, Xt, Xt–1, …) = 0

• Error terms supposed to be uncorrelated with future X’s

• This is false in the weather/OJ data

• The error term is the difference between OJ price and its population prediction based on r=18 months of weather.


population prediction based on r=18 months of weather.

• One cause of the error is that traders in OJ futures use forecasts of weather

• So if a cold winter is forecasted, traders would drive up OJ futures price, which would exceed its predicted value based on the population regression … error term ut is positive

• Now, if forecast is accurate, future weather will be cold & fdd>0 (Xt+1 positve)

• Thus cov(ut, Xt+1) > 0, violating strict exogeneity

Orange Juice Prices and Cold Weather

What is the dynamic causal effect of increase in FDD on OJ prices?

%ChgPt = β0 + β1FDDt + … + βr+1FDDt–r + ut

•What r to use? How about r = 18 months? (Guess)


•We use the rule of thumb for the Newey-West truncation:

m = .75×6121/3 = 6.4 ≅ 7

Digression: Computation of cumulative multipliers and their standard errors

The cumulative multipliers can be computed by estimating the distributed lag model, then adding up the coefficients.

However, you should also compute standard errors for the cumulative multipliers. Though this can be done directly from the distributed lag model, some modifications are required.


Because cumulative multipliers are linear combinations of regression coefficients, the methods of Section 7.3 can be applied to compute their standard errors.

Computing cumulative multipliers, ctd.

A trick in Section 7.3 is to rewrite the regression so that the

coefficients in the rewritten regression are the coefficients of

interest – here, the cumulative multipliers.

Example: Rewrite the distributed lag model with 1 lag:


Yt = β0 + β1Xt + β2Xt–1 + ut= β0 + β1Xt – β1Xt–1 + β1Xt–1 + β2Xt–1 + ut= β0 + β1(Xt –Xt–1) + (β1 + β2)Xt–1 + ut

or

Yt = β0 + β1∆Xt + (β1+ β2) Xt–1 + ut


So, let W1t = ∆Xt and W2t = Xt–1 and estimate the regression,

Yt = β0 + δ1 W1t + δ2W2t + ui

Then

δ1= β1 = impact effect


δ1= β1 = impact effect

δ2= β1 + β2 = the first cumulative multiplier

Fact: The (HAC) standard errors on δ1 and δ2 are the standard errors for the two cumulative multipliers.


In general, the ADL model can be rewritten as,

Yt = δ0 + δ1∆Xt + δ2∆Xt–1 + … + δq–1∆Xt–q+1 + δqXt–q + utwhere

δ1 = β1


1 1

δ2 = β1 + β2

δ3 = β1 + β2 + β3

…

δq = β1 + β2 + … + βq

Fact: Cumulative multipliers and their HAC SEs can be computed directly using this transformed regression

Are the OJ dynamic effects stable?

Recall from Section 14.7 that we can test for stability of time series regression coefficients using the QLR statistic. So, we can compute QLR for regression (1) in Table 15.1:

• Use HAC SEs

• How specifically would you compute the Chow statistic?

• How would you compute the QLR statistic?


• How would you compute the QLR statistic?

• d.f. q=20 (fdd+18 lags+intercept), 1% critical value = 2.43

• Result: QLR = 21.19.

This significant at 1% level – reject constancy of coeff’s

• How to interpret the result substantively? Estimate the dynamic multipliers on subsamples and see how they have changed over time…

OJ: Do the breaks matter substantively?


The cumulative effect of a FDD has declined over time. Why? Orange groves moved south, so northern frosts less damaging:


Fact: After losing many trees to freezes in northern Florida, Florida orange growers planted farther south.

Is Exogeneity Plausible? Some Examples

If X is exogenous (and ass’ns 2-4 hold), then a distributed lag model provides consistent estimators of dynamic causal effects.

As in multiple regression with cross-sectional data, we must think critically about whether X is exogenous in any application:


• is X exogenous, i.e. E(ut|Xt, Xt–1, …) = 0?

• is X strictly exogenous, i.e. E(ut|…, Xt+1, Xt, Xt–1, …) = 0?

In the following, is exogeneity plausible?

1. Y = OJ prices, X = FDD in Orlando

2. Y = Australian exports, X = US GDP (simultaneous causality, but AU economy small relative to US)

3. Y = EU exports, X = US GDP (EU economy not small relative to US)


relative to US)

4. Y = US rate of inflation, X = percentage change in world oil prices (as set by OPEC) (endogenous effect of OPEC oil price increase on inflation)

5. Y = GDP growth, X =Federal Funds rate (endogenous effect of monetary policy on output growth)

6. Y = change in the rate of inflation, X = unemployment rate on inflation (the Phillips curve)

Exogeneity, ctd.

• Evaluate exogeneity & strict exogeneity, case by case

• Exogeneity is often not plausible in time series data because of simultaneous causality

• Strict exogeneity is rarely plausible in time series data because of feedbacks


Estimation of Dynamic Causal Effects: Summary

• In theory, dynamic causal effects are measurable using a randomized controlled experiment with repeated measurements over time.

• In practice, when X is exogenous, we can consistently estimate dynamic causal effects by distributed lag regression

• If u is serially correlated, conventional OLS SEs are


• If u is serially correlated, conventional OLS SEs are incorrect; we use HAC SEs

• Think carefully in application as to whether X is exogenous

chapter 15: estimation of dynamic causal effects · is not the usual one from cross-sectional...

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