Several articles report impulse responses from policy shocks to exchange rates that never have a significant change in sign and converge to zero. Most claim that such impulse responses support some form of Dornbusch or delayed overshooting. This article shows that such impulse response functions reject overshooting from policy shocks to exchange rates. It also shows that, without additional information, such impulse responses provide no credible evidence for or against Dornbusch or delayed overshooting; that is overshooting from the policy variable itself to the exchange rate. Finally it shows that the one article that provides enough information for an appropriate test of such overshooting rejects it.
With the exceptions of [
Citation, Interval | Currencies | Confidence Interval |
---|---|---|
Eichenbaum & Evans (1995), 1974:01-1990:05 [ | 6 | ±1SD |
Grilli & Roubini (1996), 1974:12-1991:12 [ | 6 | ±1SD |
Cushman & Zha (1997), 1974-1993 [ | 1 | ±1SD |
Kim & Roubini (2000), 1974:07-1992:05 [ | 6 | ±1SD |
Kalyvitis & Michaelides (2001), 1975:01-1996:12 [ | 5 | 95% |
Faust & Rogers (2003), 1974:01-1997:12 [ | 2 | 68% |
Kim (2003), 1974:01-1996:12 [ | 1(TW) | 90% |
Jang & Ogaki (2004), 1974:01-1990:05 [ | 1 | ±1SD |
Kim (2005), 1975:01-2002:02 [ | 1 | 90% |
Scholl & Uhlig (2008), 1975:07-2002:07 [ | 4 | ±1SD |
Bjørnland (2009), 1981:I-2004:IV [ | 4 | ±1SD |
Landry (2009), 1974:I-2005:IV [ | 1(PW) | 90% |
Bouakez & Normandin (2010), 1982:11-2004:10 [ | 6 | 68% |
Heinlein & Krolzig (2012), 1972I-2009II [ | 1 | 95% |
Barnet et al. (2016), 2000:01-2008:1 [ | 1 | 68% |
Kim et al. (2017), 1981:1-2007:7 [ | 14(AG) | 68% |
Kim & Lim (2018), Approx. 1992:10-2014:9 [ | 4 | 68% |
Notes: TW: Trade weighted exchange rate; PW: Population weighted exchange rate; AG: Aggregated.
[
The models in
They also are particularly susceptible to specification search. As [
When someone submits a paper to a journal that includes estimating a model, they implicitly certify that the model makes economic sense and that the econometrics is appropriate, e.g., there is no specification search. This journal, like most, has an ethical code that prohibits the submission of articles that use data fraudulently. When a journal publishes an article, peer review implicitly re-certifies the paper. This article takes that certification as valid. It assumes that the models in
There are at least three additional problems with the articles that claim to find evidence of overshooting: 1) Unlike the Dornbusch overshooting model, no article provides a benchmark that shows what the volatility of exchange rates would be without overshooting. Without a benchmark, unless one is willing to attribute all exchange-rate volatility created by policy shocks to overshooting, there is no way to measure how much, if any, of the total volatility in exchange rates is due to overshooting. 2) Impulse responses from policy shocks, which are I(0), to exchange rates, which are I(1), reported in
The next section reviews impulse and step responses and how they relate to overshooting in a framework like the overshooting model in [
Section 4 shows that, when one excludes these conditions, impulse responses from policy shocks to exchange rates, and their corresponding step responses, are, without more information, essentially useless as tests for overshooting from policy variables themselves to exchange rates.
This section first reviews impulse and step responses.4 It then takes up the relation between those responses and overshooting. Equations (1) and (2) describe a simple discrete version of the deterministic Dornbusch model in [
s(t), p(t) and m(t) represent logs of exchange rates, price levels and money respectively. Prices depend on money while exchange rates depend on prices and money. Money in the Dornbusch overshooting model is not just econometrically exogenous, it is determined outside the model. Throughout this section we assume that m(t) is determined outside the model. We drop that assumption later.
p ( t ) = β 1 m ( t − 1 ) + β 2 m ( t − 2 ) + β 3 m ( t − 3 ) (1)
s ( t ) = a 0 p ( t ) + b 0 m ( t ) + b 1 m ( t − 1 ) + b 2 m ( t − 2 ) + b 3 m ( t − 3 ) + b 4 m ( t − 4 ) + b 5 m ( t − 5 ) (2)
with all 0 ≤ β i < 1.0 and their sum equal to 1.0, prices respond gradually to money and the quantity theory holds in the long run as in [
We begin with an impulse response function where the input is m(t) and the output is s(t).
s ( t ) = h m , s ( L ) m ( t ) (3)
In general, h m , s ( L ) = b m ( L ) / a s ( L ) where as(L) and bm(L) are polynomials in the lag operator L. Using (1) and (2) as an example, h m , s ( L ) = b 0 + ( b 1 + a 0 β 1 ) L + ( b 2 + a 0 β 2 ) L 2 + ( b 3 + a 0 β 3 ) L 3 + b 4 L 4 + b 5 L 5 .
Discrete impulse response functions like hm,s(L) describe how “outputs” like s(t) respond to a unit pulse in “inputs” like m(t). A unit pulse is zero for all t before t = 0, equals 1.0 when t = 0, and is zero for all subsequent t. There is often an implicit assumption that, before t = 0, both s(t) and m(t) have been in a steady state equilibrium with s(t) and m(t) equal to zero.
With a typical inverted “U” hm,s(L) like 0.1 + 0.3 L + 0.6 L 2 + 0.3 L 3 + 0.075 L 4 + 0.025 L 5 , a unit pulse in m(t) produces the following s(t): s(−1) = 0, s(0) = 0.1, s(1) = 0.3, s(2) = 0.6, s(3) =0.3, s(4) = 0.075, s(5) = 0.025, with all subsequent s(t) equal to zero.
Discrete step response functions describe how “outputs” like s(t) respond to a unit step in “inputs” like m(t). A unit step is zero for all t before t = 0 and equals 1.0 for t = 0 and all subsequent t. Once again there often is an implicit assumption that before the unit step the system is in equilibrium with s(t) and m(t) equal to zero. When Dornbusch describes overshooting in [
Step response functions are essentially dynamic multipliers. No economist would dream of describing how income responds over time to autonomous investment by using an impulse response function. They would use the income multiplier with respect to investment, i.e. the step response from autonomous investment to income.
If hm,s(L) is the impulse response from m(t) to s(t), then the corresponding step response gm,s(L) is the sum of that impulse response. That is g m , s ( L N ) = ∑ K = 0 N h m , s ( L K ) or g m , s ( L ) = h m , s ( L ) / Δ . Looked at from the point of view of the step response, h m , s ( L ) = Δ g m , s ( L ) . An impulse response function is the change in the corresponding step response function.
With hm,s(L) the inverted “U” of 0.1 + 0.3 L + 0.6 L 2 + 0.3 L 3 + 0.075 L 4 + 0.025 L 5 , the corresponding step response or gm,s(L) is 0.1 + 0.4 L + 1.0 L 2 + 1.3 L 3 + 1.375 L 4 + 1.4 L 5 + ⋯ + 1.4 L N . A unit step in m(t) produces the following s(t): s(−1) = 0.0, s(0) = 0.1, s(1) = 0.4, s(2) = 1.0, s(3) = 1.3, s(4) = 1.375, s(5) = 1.4 with all subsequent s(t) equal to 1.4.
Dornbusch uses a step response to describe overshooting for good reason; the relationship between impulse responses and overshooting is tenuous. Overshooting in his context is normally defined, and best discussed, in terms of step responses, not impulse responses. This article uses the following simple definition of generic overshooting that assumes a positive response: There is overshooting when some transient response to a unit step input is greater than the steady-state response.5 This is the definition implicit in
Our simple definition of overshooting defines the relation between impulse responses and overshooting. If there is overshooting, the corresponding impulse response must change sign. If it does not change sign, there is no overshooting. But a change in the sign of an impulse response does not imply overshooting. A change in the sign of the corresponding impulse response is a necessary, but not sufficient, condition for overshooting from the input to the output.
[
[
We assume that delayed overshooting is the same as “Dornbusch” overshooting except that the maximum transient response is after impact; how long after is unclear.
Continuing with our simple Dornbusch model,
The solid gm,s(L) in
The steady state response of 1.0 for this gm,s(L) provides a benchmark for measuring the amount of overshooting. For the solid step response in
The dashed gm,s(L) labeled “Delayed” in
We will return to the dotted gm,s(L) labeled Inverted “U” in
corresponding step response. These impulse responses for Dornbusch or delayed overshooting look nothing like the “U” or inverted “U” shaped impulse responses reported in
Like estimated impulse responses from “policy shocks” to exchange rates reported in
Although impulse responses from policy shocks to exchange rates in
At the beginning of the VAR overshooting literature, [
v ( t ) = ζ ( Ω t ) + e ( t ) (4)
The literature calls a unit pulse in e(t) a “policy shock” or an “innovation” in monetary policy. But giving it those names does not change what it is, simply the error tem in a regression. Note that e(t) must have the same dimension as v(t). It cannot be interpreted as the change in v(t).
When articles in
After listing some orthogonal conditions and caveats, [
Unfortunately [
Contrary to claims made by almost all the subsequent overshooting literature, [
Step responses from policy variables themselves to exchange rates provide the best way to test for some form of Dornbusch overshooting. However there are two special conditions where VAR impulse response functions alone can provide useful information. One condition is when an impulse response from a policy shock e(t) to a policy variable v(t), i.e. h e , v e ( L ) , equals 1.0. In that case, a unit pulse in e(t) produces a unit pulse in the policy variable v(t). As a result, the impulse response from e(t) to s(t) can be interpreted as the impulse response from the policy variable v(t) to the exchange rate s(t) or h v e , s e ( L ) . In that case ve(t) is effectively determined outside the model and the corresponding step response, g v e , s e ( L ) , provides a test for overshooting.
Some reported impulse responses from policy shocks to policy variables are close to 1.0 and a unit pulse in e(t) would produce something close to a unit pulse in the policy variable. But corresponding g v e , s e ( L ) reject overshooting because no transient response is greater than the steady-state response.
The other condition is when h e , v e ( L ) equals 1/Δ. In that case, a unit pulse in the policy shock e(t) produces a unit step in the policy variable v(t). In this special case, the impulse response from e(t) to s(t), h e , s e ( L ) , can be interpreted as the step response from the policy variable to the exchange rate or g v e , s e ( L ) . But this condition is inconsistent with the evidence. Reported h e , v e ( L ) in articles claiming to support some version of Dornbusch overshooting converge to something that is not statistically different from zero, usually within a few months. See for example
As pointed out above, estimated impulse response functions from policy shocks to exchange rates that have no significant change in sign reject overshooting. We now consider all of the possible ways that we can think of for how such h e , s e ( L ) might be misinterpreted as support for overshooting. If anyone can suggest a valid interpretation, we would like to know what it is.
Chris Sims pointed out to us that, if h e , v e ( L ) rise over time and converge to some value significantly greater than zero in the steady state, then it might be possible to interpret h e , s e ( L ) as the response of s(t) to something like a unit step in ve(t). In that case the typical inverted “U” shaped h e , s e ( L ) found in the literature might imply delayed overshooting from the policy variable to the exchange rate. But this possibility is inconsistent with the evidence. Essentially all reported h e , v e ( L ) converge to something that is not significantly different from zero, usually within a few months.
There is a strong possibility that several articles interpret h e , s e ( L ) as though they were g e , s e ( L ) . For example, [
If they were describing a step response from e(t) to s(t), i.e. g e , s e ( L ) , it would support overshooting from e(t) to s(t). But they are describing an impulse response from a policy shock to an exchange rate, i.e. an h e , s e ( L ) , that does not have a significant change in sign. Such impulse response functions reject overshooting from e(t) to s(t).
Another possibility is that h e , s e ( L ) estimated by VAR are special. They somehow can be interpreted as g e , s e ( L ) . We use RATS to debunk that possibility.
We use IMPULSE.PRG from RATS to estimate the following three equation model describing Dornbusch overshooting using a Choleski decomposition. To keep the model relatively simple, as in [
m ( t ) = e ( t ) (5)
p ( t ) = 0.1 m ( t ) + 0.6 m ( t − 1 ) + 0.3 m ( t − 2 ) + e 1 ( 2 ) (6)
s ( t ) = 1.4 m ( t ) − 0.8 m ( t − 1 ) − 0.45 m ( t − 2 ) − 0.075 m ( t − 3 ) − 0.05 m ( t − 4 ) − 0.025 m ( t − 5 ) + p ( t ) + e 2 ( t ) (7)
where e(t), e1(t) and e2(t) are orthogonal white noise error terms by construction. Ignoring the error terms, the deterministic g m , s e ( L ) for this model produces the step response for Dornbusch overshooting in
Replacing (6) and (7) with (8) and (9) produces the delayed overshooting in
p ( t ) = 0.1 m ( t ) + 0.6 m ( t − 1 ) + 0.3 m ( t − 2 ) + e 1 ( 2 ) (8)
s ( t ) = 1.0 m ( t ) − 0.3 m ( t − 1 ) − 0.2 ( t − 2 ) − 0.3 m ( t − 3 ) − 0.15 m ( t − 4 ) − 0.05 m ( t − 5 ) + p ( t ) + e 2 ( t ) (9)
There also is an example that produces the inverted “U” described above.
VAR impulse responses are conventional impulse response functions. One cannot interpret a VAR impulse response from a policy shock to an exchange rate as though it were a step response from e(t) to s(t).
This misinterpretation is similar to the previous one. Somehow a unit root in s(t) transforms h e , s e ( L ) into g e , s e ( L ) . We continue to assume that m(t) is stationary for two reasons: First common policy variables like short-term interest rates, short-term interest rate differentials and NBRX are likely to be stationary. Second reported h e , v e ( L ) imply that ve(t) are stationary because the h e , v e ( L ) converge to zero.
Equations (10) to (12) describe a simple VAR model with Dornbusch overshooting where m(t) is stationary and determined outside the model, but s(t) has a unit root because p(t) has a unit root.
m ( t ) = e ( t ) (10)
p ( t ) = 1.0 p ( t − 1 ) + e 1 ( 2 ) (11)
s ( t ) = 1.5 m ( t ) − 0.2 m ( t − 1 ) − 0.15 ( t − 2 ) − 0.075 m ( t − 3 ) − 0.05 m ( t − 4 ) − 0.025 m ( t − 5 ) + p ( t ) + e 2 ( t ) (12)
Changing Equation (12) to Equation (13) changes the model to one with delayed overshooting.
s ( t ) = 1.1 m ( t ) + 0.3 m ( t − 1 ) + 0.1 m ( t − 2 ) − 0.3 m ( t − 3 ) − 0.15 m ( t − 4 ) − 0.05 m ( t − 5 ) + p ( t ) + e 2 ( t ) (13)
Changing Equation (13) to Equation (14) changes the model to one with an inverted “U”.
s ( t ) = 0.1 m ( t ) + 0.3 m ( t − 1 ) + 0.6 m ( t − 2 ) + 0.3 m ( t − 3 ) + 0.15 m ( t − 4 ) + 0.025 m ( t − 5 ) + p ( t ) + e 2 ( t ) (14)
There is generic overshooting when some transient response to a unit step in the input is greater than the steady-state response of the output. It is possible that some articles implicitly redefined overshooting. They replace the unit step with a unit pulse. This redefinition violates the definition implicit in [
In models with rational expectations, white noise errors or “innovations” represent “information” that changes the output of the model permanently. Although the authors in
While unit steps in e(t) might be interpreted as “information”, unit pulses are difficult to interpret as “information”. As long as the relationship described by h e , s e ( L ) is stable, a unit pulse in e(t) does not change the steady state value of s(t).8 The new steady state must be the same as the original steady state.
Some articles probably interpret “policy shocks”, or e(t), as changes in the policy variable itself, or Δv(t). Misinterpreting e(t) as Δv(t) would help explain why so many articles in
If one could interpret e(t) as Δv(t), then one could write s e ( t ) = h e , s e ( L ) e ( t ) as s e ( t ) = h e , s e ( L ) Δ v ( t ) , which would imply that the impulse response from e(t) to s(t), h e , s e ( L ) , was the step response from v(t) to se(t), i.e. g v , s e ( L ) . In that case the inverted “U” h e , s e ( L ) reported in
To summarize, articles in
The previous section assumes that when articles in
Without more information, the h e , s e ( L ) reported in
We illustrate this point first by showing how h e , s e ( L ) can appear to imply delayed overshooting from e(t) to s(t) when there is no overshooting from m(t) to s(t). Then we show how h e , s e ( L ) can reject overshooting from e(t) to s(t) when there is delayed overshooting from m(t) to s(t). In both cases the culprit is the “endogeneity” of the policy variable.
For simplicity we use the model described by Equations (15) to (17).
m ( t ) = b 1 m ( t − 1 ) + b 2 m ( t − 2 ) + b 3 m ( t − 3 ) + e ( t ) (15)
p ( t ) = β 0 m ( t ) + β 1 m ( t − 1 ) + β 2 m ( t − 2 ) + e 1 ( t ) (16)
s ( t ) = a 0 p ( t ) + γ 0 m ( t ) + γ 1 m ( t − 1 ) + γ 2 m ( t − 2 ) + γ 3 m ( t − 3 ) + γ 4 m ( t − 4 ) + γ 5 m ( t − 5 ) + e 2 ( t ) (17)
Equation (15) determines the extent of the “endogeneity. With b1, b2 and b3, all zero, h e , m e ( L ) equals 1 and m(t) is effectively determined outside the model. Otherwise, unlike the Dornbusch overshooting model, m(t) is determined at least partly within the model. h e , m e ( L ) and g e , m e ( L ) describe the impulse and step responses from e(t) to m(t) implied by Equation (15). As before, h e , s e ( L ) is the impulse response from the policy shock to the exchange rate and g e , s e ( L ) is the corresponding step response.
Equations (16) and (17) determine whether or not there is Dornbusch or delayed overshooting. That is whether or not a unit step in m(t) produces an impact response, or some other transient step response from m(t) to s(t), that is greater than the steady state response and converges to something that is above zero. We use g m , s m ( L ) to describe that step response and h m , s m ( L ) to describe the corresponding impulse response.
Equations (18) to (20) provide a numerical example where there is no overshooting from m(t) to s(t) because no transient g m , s m ( L ) is greater than the steady state response, but the g e , s e ( L ) implies delayed overshooting from e(t) to s(t) because there is overshooting from e(t) to m(t).
m ( t ) = 0.5 m ( t − 1 ) − 0.5 m ( t − 2 ) + e ( t ) (18)
p ( t ) = 0.1 m ( t ) + 0.6 m ( t − 1 ) + 0.3 m ( t − 2 ) + e 1 ( t ) (19)
s ( t ) = p ( t ) + 0.9 m ( t ) − 0.6 m ( t − 1 ) − 0.3 m ( t − 2 ) + e 2 ( t ) (20)
There is no overshooting from m(t) to s(t) because the g m , s m ( L ) equals 1.0 for all LN. No transient step response is greater than the steady state step response. But this response of s(t) to a unit step in m(t) is more than offset by overshooting from e(t) to m(t) where the g e , m e ( L ) is 1.0 , 1.5 L , 1.0 L 2 , ⋯ , 1.0 L N . That combination of g e , m e ( L ) and g m , s m ( L ) produces the following g e , s e ( L ) : 1.0 , 1.5 L , 1.0 L 2 , ⋯ , 1.0 L N . The maximum transient step response is after impact and it is greater than the steady-state response. This g e , s e ( L ) implies delayed overshooting from e(t) to s(t) and the corresponding h e , s e ( L ) is consistent with that interpretation because it changes sign. But there is no Dornbusch or delayed overshooting from m(t) to s(t), only an endogenous m(t).
Equations (21) to (23) illustrate the opposite possibility; there is delayed overshooting from m(t) to s(t), but the g e , s e ( L ) shows no evidence of overshooting from e(t) to s(t) because the undershooting from e(t) to m(t) hides the overshooting from m(t) to s(t).9
m ( t ) = 0.3 m ( t − 1 ) + 0.2 m ( t − 2 ) + 0.1 m ( t − 3 ) + e ( t ) (21)
p ( t ) = 0.1 m ( t ) + 0.6 m ( t − 1 ) + 0.3 m ( t − 2 ) + e 1 ( t ) (22)
s ( t ) = p ( t ) + 1.0 m ( t ) − 0.3 m ( t − 1 ) − 0.2 m ( t − 2 ) − 0.3 m ( t − 3 ) − 0.15 m ( t − 4 ) − 0.05 m ( t − 5 ) + e 2 ( t ) (23)
There is delayed overshooting because the g m , s m ( L ) in this model is 1.1, 1.4L, 1.5L2, 1.2L3, 1.05L4 from where it converges 1.0. But the undershooting from e(t) to m(t) overwhelms that overshooting and the g e , s e ( L ) is 1.1, 1.7L, 2.2L2, 2.3L3 from where it converges to 2.5. There is no overshooting from e(t) to s(t) because no transient step response is greater than the steady-state response.
As this section illustrates, without additional information, impulse responses from policy shocks to exchange rates tell us nothing useful about overshooting from policy variables to exchange rates. Unfortunately the VAR literature often seems to draw inappropriate conclusions about Dornbusch or delayed overshooting based solely on impulse responses from policy shocks to exchange rates that tell us nothing about such overshooting.
Only one article in
Heinlein and Krolzig estimate a fully identified model with five variables: 1) an output gap differential (yd), 2) an inflation gap differential (πd), 3) a three month T bill rate differential (id), 4) a 10 year bond rate differential (rd) and 5) the dollar price of sterling (e) where id is the policy variable and all differentials are U.K. minus U.S. To avoid complicating the notation, we refer to their policy variable as v(t), their policy shock as e(t) and their exchange rate as s(t).
They avoid the problems created by unit roots by estimating the model in first differences. To be consistent with the other literature, we retrieve levels by the simple expedient of adding the lagged value of the dependent variable to both sides of their equations. For example, if they estimate Δ y ( t ) = − α y ( t − 1 ) + β x ( t ) + x ( t − 1 ) , we convert it to y ( t ) = ( 1 − α ) y ( t − 1 ) + β x ( t ) + x ( t − 1 ) .
Estimates of their PSVECM model, which is their preferred model, provide the information needed to construct a step response from the policy variable to the exchange rate where their policy variable is determined outside the model.
Like other articles in
The solid impulse response labeled “e(t)” in
The solid response in
The dashed impulse response in
[
Articles in
policy shocks to exchange rates that never have a significant change in sign and converge to zero. Our first and most important point is that, taking them as valid, such impulse response functions clearly reject overshooting from policy shocks to exchange rates. They imply corresponding step response functions where no transient response is greater than the steady state response. In other words, a permanent, rather than temporary, increase in what is called the “policy shock” would not cause the exchange rate to rise by more in the short run than in the long run.
Our second point is that the impulse responses in
Put succinctly, the evidence in
This article concentrates on the misinterpretation of impulse response functions in testing for Dornbusch and delayed overshooting; future research on Dornbusch and delayed overshooting needs to use a wider variety of econometric techniques and needs to evaluate impulse responses more carefully.
If this article is correct, then the articles in
I want to thank Tom Doan at Estima, Chris Sims, an anonymous referee and particularly Michael Pippenger for their comments and suggestions. Any remaining errors are of course mine.
The author declares no conflicts of interest regarding the publication of this paper.
Pippenger, J. (2019) Testing for Dornbusch and Delayed Overshooting: Setting the Record Straight. Theoretical Economics Letters, 9, 1489-1506. https://doi.org/10.4236/tel.2019.95096