ECON 0024: Economic Policy Analysis

Estimating the Effects of Monetary Policy

Daniel Lewis
University College London

December 3, 2022
Review

Last time, we learned about the goals and tools of monetary policy and how
the economy is monitored.

• Central banks set interest rates to adjust the supply of money in the
economy.
• They either set interest rates directly or buy/sell securities to affect them.

• Normally, rely on interest rates filtering through the economy via financial
institutions.
• Desperate times call for unconventional policy: forward guidance,
quantitative easing.
• Central banks currently face the possibility of stagflation and new threats
to their independence.
• Monitoring the state of the economy is a challenge in its own right.
Today

So now you know the goals and tools of monetary policy. Does it work?

• We cannot conduct randomised controlled experiments (how???).

• We cannot rely on natural experiments either (what is the “control”?).

• Macroeconomy is also a complicated web of interdependent variables.

• Instead, need clever econometric strategies/assumptions.

• We can never just regress outcomes on changes in interest rates (why?).

• Also need to undertand whether unconventional policy works – as

well/better/worse than conventional?
• This lecture will draw heavily on material from ECON 0019.
Autoregressive models and more

Recall the AR(p) model

yt = αy + β1 yt−1 + β2 yt−2 + . . . + βp yt−p + uy ,t .

Let yt be inflation, πt – doesn’t just depend on past inflation, but
unemployment, interest rates, output, etc. – a distributed lag model

πt = απ + β1,π πt−1 + . . . + βp,π πt−p

+β1,p Ut−1 + . . . + βp,U Ut−p
+β1,r rt−1 + . . . + βp,r rt−p + uπ,t .

We can rewrite this as

 
p
X πt−l
πt = απ + (βl,π , βl,U , βl,r ) Ut−l  + uπ,t .
l=1 rt−l
Towards vector auto-regression
Of course, we could similarly estimate models for Ut and rt !
   
p
X
πt−l Xp

πt−l
Ut = αU + ρl,π , ρl,U , ρl,r Ut−l  + uU,t , rt = αr + γl,π , γl,U , γl,r Ut−l  + ur ,t .
l=1 rt−l l=1 rt−l

Now we have three separate models: one each for inflation, unemployment, and
interest rates.
Together, they allow us to trace the effect of a change in one variable through
the whole economy over time!
More parsimonious (but completely equivalent!) to combine them into a single
model, a vector auto-regression (VAR):
p
X
Yt = |{z}
α + Bl Yt−l + ut ,
|{z} |{z} |{z} |{z}
3×1 3×1 l=1 3×1
3×3 3×1

where
    
πt uπ,t βl,π βl,U βl,r
Yt = Ut  , ut = uU,t  , Bl = ρl,π ρl,U ρl,r  .
rt ur ,t γl,π γl,U γl,r
VARs and forecasting

Simple VARs like this one are useful for forecasting.

• Each of the n = 3 equations gives us a prediction for one of the variables
of interest given all past data.
• But they also estimate the change in πt associated with a change in rt−l .
• They account for all of the indirect relationships, filtering through the
other variables over time:
∂πt
̸ Bl,13
=
∂rt−l
∂πt ∂πt−1 ∂Ut−1 ∂rt−1
= B1,11 + B1,12 + B1,13 + . . . + Bl,13
∂rt−l ∂rt−l ∂rt−l ∂rt−l
Impulse response functions

The change in one variable associated with a change in another at some lag is
called an impulse response function (IRF): cornerstone of empirical macro.
Conceptually, think of these as response of Yt+h to a change in uj,t – why?
IRF at horizon h can be computed recursively from knowledge of B1 , . . . , Bp :
∂Yt
Φ0j = = ιj , ιj is a vector with 1 in j entry and zeros o.w.
∂uj,t
min(p,h)
X
Φhj = Bk Φh−k
j
k=1
A conceptual framework for causal analysis

So far we have considered the time series analog of correlations. What are the
requirements for a causal estimate of a change in macroeconomic policy?
We want to study responses to changes in policy that are unpredictable:
• If it’s predictable, you’re estimating effect of whatever predicts it!
• If it’s anticipated, then rational agents have already responded!
• A change that is predicted or caused by another change is endogenous.
• A change that is unpredictable and not a causal result of something else
is exogenous.
A movement in a macroeconomic series that is exogenous a structural shock.
Want to study structural shocks as as-if random variation in policy – surprises.
From residuals to shocks and the SVAR model

The vector ut of VAR residuals is unpredictable (why?) but ut are not shocks.
• They are cross-correlated – exogenous shocks should be uncorrelated
(why?).
• So residuals ut are endogenous: represent the effects of underlying
(latent) shocks in the economy.
• The Structural VAR (SVAR) model expresses the residuals as linear
combinations of n unobserved uncorrelated shocks, ϵt :

ut = Aϵt , E [ϵt ] = 0, E [ϵt ϵ′t ] = In , E [ϵt ϵs ] = 0, t ̸= s

p
X
Yt = α+ Bl Yt−l + Aϵt .
l=1

The full-rank matrix A is the contemporaneous response matrix. Aij is

the effect of a unit shock j on variable i. Interpreted causally.
How to recover the shocks?

How do we estimate A? Or, equivalently, εt ?

We cannot, without further assumptions!

E [ut ] = AE [ϵt ] = A × 0 = 0 tells us nothing!

E [ut ut′ ] = AE [ϵt ϵ′t ]A′ = AA′ is (n2 + n)/2 equations in n2 unknowns!

Need an additional (n2 − n)/2 restrictions to find a unique solution.

The simplest approach is recursive/timing restrictions: restrict which variables
can respond to which shocks contemporaneously.
Result:  
A11 0 0
A = A21 A22 0 
A31 A32 A33
With ordering Yt = (πt , Ut , rt )′ , inflation responds only to price shock
contemporaneously, unemployment to price and labour market shocks, and
interest rate responds to price, labour market, and monetary policy shocks.
Structural impulse responses

Under these restrictions, A is the unique lower triangular (Cholesky) factor of

E [ut ut′ ] = cov (ut ) = AA′ .

Once we have an estimate for A, we can compute structural impulse

responses: the dynamic causal effects of monetary policy.
Regular impulse responses measure the effect of unit change in uj,t on Yt+h .
A measures the effects of unit structural shock on ut .
So use the product rule to compute effect of structural shocks on Yt+h :

∂Yt+h ∂Yt+h ∂ut

Θhj = = = Φh A·j
∂ϵj,t ∂ut′ ∂ϵj,t
Some alternatives

Timing assumptions are strong assumptions – may want others.

• Are the restrictions plausible in monthly data? Quarterly?

• Long-run restrictions are the opposite: assume instead effects of some

shocks die out at long horizons.
• Sign restrictions are very different: if we are sure that some elements in
A have certain signs (e.g., interest rate hike lowers inflation), rule out
solutions for A that violate those restrictions. Gives a range of possible
values, not single estimate.
• Can also use instrumental variables: suppose some instrument is
correlated with monetary policy shock (but not price or labour market
shocks); then can estimate corresponding column of A.
• In practice: regress each series in ut on instrument zt by OLS. Normalize
vector of coefficients by the coefficient of ur ,t to get A·mp .
Local projections

There is an alternative way to estimate the effects of monetary policy (or other
macro policies): local projections (LPs).
• Local projections are really just direct OLS regressions to forecast some
outcome.
p
X
yt+h − yt−1 = a + λh rt + κl Xt−l + vt ,
l=1

where Xt is a set of controls (inflation, unemployment, rates).

• Generally more flexible than a VAR.

• The IRF at horizon h is just λh .

• But this is just the response to the part of rt unpredictable by

Xt−1 , . . . , Xt−p . Not causal yet (why?).
Local projections with instrumental variables

There are many approaches to obtain causal responses with SVARs, but LPs
almost always use instrumental variables.

• This is really easy: just like 2SLS.

• First stage: regress rt on zt and lags of Xt , predict rˆt .

• Second stage: regress (yt+h − yt−1 ) on rˆt plus controls, coefficient λh is

dynamic causal effect at horizon h.
• What types of instruments can be used for monetary policy?
1 High frequency financial data
2 “Narrative” rate changes regressed on internal CB forecasts
• In general, you should not just use the measured shocks directly as the
independent variable – why?
• Including controls is important: exogeneity and relevance.
Christiano, Eichenbaum, & Evans (1999): recursive
Uhlig (2005): sign restrictions
Romer & Romer (2004): as instrument in SVAR
Gertler Karadi (2015): instrument in SVAR
Gertler & Karadi (2015): instrument in LP
Evaluating “macroeconometric” models

What have we learned about estimating causal effects in macroeconomic

settings?

• There is no single way to estimate effects even given the same dataset.

• Results are sensitive to the restrictions/assumptions imposed.

• Typically, assumptions cannot be tested (why?).

• Since there is no “truth”, important to think about robustness: are there

findings that are relatively insensitive to alternative assumptions?
Is conventional monetary policy effective?

For one of the most important questions in empirical macro, the answers are
surprisingly tentative.
• Previously saw some examples of IRFs to monetary policy shocks.
• The price puzzle.
• Ambiguous response of real activity under sign restrictions, etc.
Summary of leading results (Ramey (2016))
What about unconventional policy?

The literature on unconventional policy is still very limited.

• For the most part, policies new starting in 2008.

• Most short-lived and time-varying – very small sample.

• The sample contains little but a massive recession.

• Early evidence shows strong effects of some forms of forward guidance

and asset purchases on financial markets.
• Even less evidence on most important macro variables: evidence that
quantitative easing was effective, but forward guidance less so.
• “QE works in practice, but not in theory”.
Interpretability of estimates

We had to make some conceptual leaps to recover “causal estimates”.

• What is a “structural shock” really?

• We hope central banks don’t set interest rates at random...

• Interest rate changes are almost entirely predicted by markets/forecasters.

• Does a central bank have to do something surprising to have any effect?

Or do anticipated changes have an impact?
• Do today’s estimates have any connection to actual changes in interest
rates following an MPC meeting?
• And that’s without thinking about all the assumptions...
Wrapping up

Estimating the effects of monetary policy is really hard – but essential for policy.

• We have strong intuition/theory for the effects, but sometimes

remarkably hard to match that in the data.
• Important: avoid wasting time/placing faith in policies that don’t work.

• Particularly relevant right now – rate hikes with persistent inflation.

• Alternatives: even more assumptions in “structural” models,

“experimental macro”.
• Next few years will feature important new data and evaluation.