Lecture 19 Flashcards
(15 cards)
Why use time series for causality?
Crucial when asking causal questions like:
- what’s the effect of interest rates on inflation?
- what happens to output when gov spending increases?
What is a dynamic causal effect?
What is the effect of a change in x today on y tomorrow, the day after, etc
- looking at how the effect evolves over time
E.g. if interest rates increase today, how does GDP respond this quarter, next quarter and further into the future?
Conceptual challenge of doing causal inference with time series data
- issue
- solution
- key assumptions
Issue:
- no Randomised controlled trial in time series, as only have one unit, can’t randomly assign treatment
Solution:
- to justify causal inference in this setup, reframe the time series as one subject, observed at many time points, sometimes receiving a treatment, sometimes not
Key assumption:
- stationarity, if the distribution doesn’t change over time, we can treat this like repeated sampling of the same subject under different treatment conditions
How to estimate dynamic causal effects in time series with a Distributed Lag model
- regressing current outcome yt on current and past values of xt
B0 - impact effect, so immediate effect of a change in xt on yt, holding previous x values fixed
B1 - 1 period lagged effect, effect of xt-1 on yt, holding other lags constant
- cumulative dynamic multiplier: total effect, e.g., 2 period CDN is B0 + B1 + B2
IRF - Impulse Response Function, a graph of how yt responds over time to a one-unit shock in xt
Why DL instead of earlier naive regression for causal reference
Naive regression only captured the contemporaneous effect, B0, whereas DL captures the dynamic causal effect - how effects unfold over multiple months
- IRF summarises this path graphically
Use naive if you think effect is instantaneous only, use Dl if you want to capture delayed and cumulative effects
Strict exogeneity vs Contemporaneous Exogeneity
Strict: E[ut|Xt,…,X1] = 0, error must be uncorrelated with all values of X, past and future
- needed for unbiased estimation of dynamic causal effects in DL models
Contemporaneous: E[ut|Xt] = 0, easier to satisfy
- with this alone, OLS is still consistent, but not unbiased in finite samples
For dynamic causal interpretation of DL coefficients,
We ideally need strict exogeneity
Assumptions for valid DL estimation
- Linearity
- Exogeneity - strict or contemporaneous
- No perfect collinearity
- Stationarity
- Weak dependence/ ergodicity
First 3 + strict gives unbiased DL
All 5 but with non strict gives consistent and asymptotically normal
Inference on cumulative multipliers
E.g., want to know how precise cumulative effect B1 + B2 is,
- var(B0 + B1) = var(B0) + var(B1) + 2Cov(B0,B1)
How to estimate cumulative multipliers directly
Yt = a + B0.xt + B1xt-1 + ut
= a + B0.xt - B0xt-1 + B0.xt-1 + B1xt-1 + ut
= a + B0(xt - xt-1) + (B0 + B1)xt-1 + ut
Then regress yt on TRI.xt and xt-1, OLS coefficients will be the impact effect and first cumulative multiplier
What the general answer?
Control in DL models, as strict exogeneity not realistic
Conditional mean independence instead
- E[ut | xt, wt-1,…, wt-l] = E[ut|wt-1,…,wt-l]
- xt is exogenous after controlling for some w’s
- breaks down for long lags, introducing bias in long-lag coefficents
What is a local projection
A way to estimate how a shock today affects an outcome in the future
E.g. want to know the effect of a shock today on GDP 1 month later?, run yt+1 - yt-1 = a1 + B1xt + controls + u1,t
- and so on for each horizon h = 0,1,2..H
- can now plot a path showing the effect over time - impulse response function
Advantages of LPs
- Direct estimation of impulse responses
- each Bb is literally the estimated response at horizon h - Flexibility with controls
- can flexibly add controls to help exogeneity - Weaker assumptions
- dont need strict exogeneity of xt across all future periods, just need Bb to be unbiased given the controls
DL models and exogeneity
Until now have used DL models with strong assumptions:
- xt is strictly exogenous
Alternatively can use ADL models:
- includes lags of yt as regressors, can sometimes give more efficient estimates, but harder to interpret causally
Tradeoff:
- DL = simple, but requires strong exogeneity assumptions
- ADL = more flexible, but introduces bias unless exogeneity is carefully handled
Practical difficulty with lags for DL
- Need enough lags of the control wt to block correlations between xt and ut
- if past x influences future w, then controlling for lags wont fully fix endogeneity
Risk: including too many controls makes longer-horizon impulse response coefficients inconsistent
