Lecture 19 Flashcards
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
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 B’s over time, plotting the estimated effect of a 1-unit shock to xt on y over future periods
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(B1 + B2) = 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
Difference between DCEs via DL models and Local Projections
DCE:
- estimate all herons at once in a single regression using lags of xt
- requires strong assumption but more efficient if model is correctly specified
- can recover impulse responses by adding coefficients up on lags
LP:
- estimate one horizon at a time by running separate regressions for each future outcome
- more flexible and robust to misspecification and can include controls
