Review of Methodologies




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Multivariate Models III

Structural VARs
Review of Methodologies:
Univariate models:

AR, MA, ARIMA models. Must be stationary to model them as ARMA.

Nonstationary models – spurious regressions. Nonstationarity tests (ADF, PP, etc.).
Multivariate models single equation models:

Stationary variables – OLS

Nonstationary variables

–cointegration: residual based nonstationarity tests, ECM.


Multivariate models: VAR

Stationary variables – OLS or UVAR in levels.

Nonstationary variables –cointegration and Johansen MLE method, VECM (CVAR). If there is more than 1 CI vector, you cannot identify them, must impose identification restrictions on .
How do we choose between UVAR, SVAR, CVAR (VECM)?


  • If the goal is inference, estimate of parameters (e.g. quantifying SR responses), must worry about nonstationarity, use VECM. If variables are I(1) and there is no CI vector, model may be misspecified, respecify it. If there is no theoretical model that provides a CI vector, then first-difference the variables. You should also use VECM if your goal is the LR implications of the model.

  • If the goal is forecasting or impulse responses (e.g., policy analysis: how variables respond to policy shocks given policy constraints, CB’s reaction function, how much i must change to to offset the rise in unemployment rate? ) no need to worry about nonstationarity. Add sufficient number of lags to remove serial correlation and make the errors I(0) and proceed to the analysis. Not adding the CI term will make you lose efficiency, but this will not affect the forecasting or the impulse responses.

    • If variables are correlated with each other, the error terms in the UVAR will be correlated across equations. To identify shocks, you must make them orthogonal  uncorrelated between equations. There are two possibilities:

  1. Use a recursive VAR (Cholesky decomposition). But this is ad hoc and results are order-dependent.

  2. Impose contemporaneous SR restrictions on SVAR on levels whether the variables are I(1) or I(0). Add enough lags to get I(0) errors. With these “identifying assumptions” , correlations can be interpreted causally. Ex: a Taylor rule sets the interest rate equal to lagged inflation and unemployment (=instrumental variable regression) and is the interest rate equation in the VAR.

  3. Impose LR restrictions (Blanchard and Quah). Then you must have at least one I(1) variable and all series in VAR must be I(0). First-difference the I(1) variable to run the VAR with the other I(0) variables (must make sure that first-differencing makes sense theoretically).

For further details on VAR, UVAR and SVAR see: Stock and Watson, Vectorautoregressions, JEP (2001).


Ref: Enders Ch.5, Favero Ch.6, Bernanke and Mihov (1998, QJE), Blanchard and Quah (1989, AER)
Recall the VAR(1) model with 2 variables:

(1) SVAR or the Primitive System

(2)

and
Or in matrix form:

(3)


More simply:

(4)


To get the VAR in standard form: multiply the equation by inverse B:

,

(5)

or:

(6)



Error terms are composites of the structural innovations from the primitive system.

(7)

Or

where


The var/covar matrix of the VAR shocks:

.
Identification

To get the effect of a structural innovation on the dependent variables we recover the parameters for the primitive system from the estimated system.

VAR: 9 parameters; SVAR: 10 parameters, underidentified.
A triangular Cholesky decomposition makes the SVAR exactly identified: impose the 0 condition on or .
Problem with this identification restrictions:


  • impulse responses depend on the ordering chosen. If the correlation between the variables is low, it doesn’t matter but unlikely in time series.

  • How do we decide on which b should we impose the restrictions? Can be very counterintuitive.

Sims (1986), Bernanke (1986) proposed using theory to derive the identifying restrictions in B:

Estimation gives the error terms e defined as so

.

Consider a VAR(1) with n variables:



where B and are (nxn) and is an (nx1) matrices.

.

To know whether we have an identified VAR or not, count #knowns and #unkowns


Exact identification: order condition

Known estimates: distinct elements of the var/covar matrix:

altogether distinct elements.
Unknown parameters: elements of B and var/cov matrix of structural shocks (pure shocks):

  • Diagonal elements of B are 1 and known, so unknown parameters in B =

  • Pure structural shock’s covariances are zero.

 altogether n unknown elements in cov matrix.

Total number of unknowns =


Additional restrictions needed to identify the system:

Necessary condition for exact identification

More formally:




Hence:

Ex: Cholesky decomposition:

All elements above the diagonal are zero, thus restricted:

i.e. additional restrictions. Thus the system becomes exactly identified.

Ex: Decomposition with a two-variable VAR(1)


t

1

2

3

4

5





1

-0.5

0

-1

0.5

0



0.5

-1

0

-0.5

1

0

Identification restrictions: n=1, because known=2(1+2)/2=3, unknown=22=4. We need one more restriction. To see this numerically:





and
We have the relation

or where .
Hence


4 equations and 4 unknowns: , , .

But actually there are 3 equations since the diagonal elements of are identical.










The 2nd and the third equations are identical. So we need another restriction to solve the relations:




(i) Cholesky decomposition:

We saw before that this leads to an exact identification. Number of knowns=2(3)/2=3.

# unknowns = 22-1=3. The restriction corresponds to a recursive system, where the most endogenous variable is placed last.
Numerically:





Since we can now identify the structural shocks:

hence
We can then trace back the structural shocks at every point in time:


t

1

2

3

4

5





1

-0.5

0

-1

0.5

0



-0.3

-0.6

0

0.3

0.6

0



(ii) Structural models with contemporaneous coefficient restriction:

ex.

We had:


Justification for the restriction:

Suppose thus and we can pin down the first structural shock:



Then we get the remaining unknowns by substituting in:








and


(iii) Symmetry restriction ex: then multiple solution

a.


b.
(iv) VAR restriction on variances of structural shocks: multiple solutions for coefficients of B.

ex:




a.
b.

(v) Model restrictions: overidentification

If #restrictions > (n2-n)/2. This does not affect the estimation of the VAR coefficients.
Steps to follow to test the significance of restrictions:


  1. Run VAR without additional restrictions and get the var-covar matrix .

  2. Impose the additional restrictions and maximize the likelihood function, get --restricted—

  3. Calculate , which will be distributed as

where R=#restrictions-(n2-n)/2. If calculated test statistics less than tabulated values, do not reject the null (of restrictions).

How did the VARs enter the monetary policy analyses?

After the collapse of confidence in large macro models to analyze policy, Sims (1980) proposed a methodology free of a structural model. He introduced VAR analysis to study the monetary transmission mechanism between money and GDP.


VAR with logs of GDP, P level and money (M1 or M2).
Choleski ordering:


Although there is no formal model, the VAR is based on a standard theoretical model :

y is exogenous, only affected by its own shocks (output equation)

p is affected contemporaneously by its own shocks but also by y shocks and it is not affected by m shocks (from an inflation equation such as PC).

m is affected by its own and the two other shocks (money demand equation).

Subsequent additions to the model:

interest rate i (Fedfund rate), e.g. Leeper, Sims and Zha (1997) estimate a level VAR with the order:




Despite the appeal of this methodology, researchers had to deal with several persistent counterintuitive results in both closed economy and open economy VAR MODELS:
Closed economy

Liquidity Puzzle: a typical finding was not to find the familiar liquidity effect (an inverse relation between a money market shock and interest rates). A money market shock increases the interest rate, instead of decreasing it.

Literature background


Early research: increases in MS reduce interest rates (Gibson 1968, QJE, Cagan and Gandolfi, 1968 AER P&P).

Research in 1980s no good news: this effect disappears in the 1970s. Tests started to focus on unanticipated changes in policy –Rational expectations revolution—.

Findings: Correlations between innovations in M and i.r. positive or insignificant; announcements of MS larger than expected associated with increases in ST i.r.

Recent literature mixed:

Findings: eliminating high frequency movements in MS and i.r. reestablished some LE for subsamples (Cochrane, 89). LE is supported if (i) there is LR identification consisting of M neutrality (Gali, 92); (ii) emphasis on bank reserves and the FF rate (Strongin, 95). But an equal number of studies show that evidence supports the a positive relation between liquidity and i.r.
Possible reasons for the puzzle:


  • If you are using growth of money supply, then a rise in M growth will increase inflationary expectations, which will push nominal rates up (Fisher equation). So you must be careful between the level and the change in the MS.

  • This shock can be interpreted as a Money demand shock. Then XD in the MM will lead to a rise in the interest rate.

  • If you are using a too wide definition of M beyond the Fed’s, it may involve the behavior of the banking sector. So use a narrow monetary aggregate.


Price Puzzle: a positive interest rate shock (contractionary monetary policy) leads to persistent rise in the price level. A tightening of MP generally is expected to reduce the P level, not increase it. Result holds for other countries as well and is especially pronounced if short-term interest rate is taken to be the policy instrument rather than a monetary aggregate.
Possible reasons for puzzle:

  • Misspecification of the model. Sims (1992, EER): a factor may be missing in the reaction function of the CB. If the Fed is reacting to a leading indicator (e.g. commodity price index) and increasing the rates, then we observe the simultaneous rise in i.r. and P.

Christiano, Eichenbaum, Evans (1996), Sims (1996): price puzzle disappears.

Hanson (2004, JME) compares various commodity P indices: (i) little correlation between those indices that forecast best the P inflation and those that resolve the puzzle; (ii) the indices that solve the puzzle best are broad ones or financial data as opposed to individual commodity price indices.



  • Puzzle may be due to using detrended output instead of output gap (Giordani, 2004, JME). The monetary policy shock gets contaminated by other shocks if detrended output is used. The VAR becomes triangular if output gap is taken.


Non-neutrality of money in the LR: a wide consensus in the literature is that money has real effects in the short run but it is neutral in the long run, and changes in the MS do not have a LR effect on the real variables such as output, employment, real interest rates, real money balances.

Both old and recent studies including VAR analysis (1960s-1990s) finds that changes in MP lead to long, protracted response in real variables.



Open economy

In addition to the usual closed economy variables, the crucial variable included to the list is the bilateral exchange rate along with the foreign variables.


Forward-discount puzzle. According to the Uncovered Interest Rate Parity: , where i=domestic rate, i*=foreign rate, e=spot exchange rate (defined as the dollar price of the foreign currency, $/for.curr; a rise means depreciation of the dollar) and expected future spot rate, approximated as the forward rate or the long-term value of e. A positive innovation in i should lead to an impact appreciation (with a fall in e>fall in the future rate, thus expected depreciation since the term in parentheses will go up, also called forward discount on the dollar if positive) followed by spot depreciation over time as i will converge towards i*.

IRs show that a restrictive US monetary shock (interest rate shock) continuously appreciates the dollar, and increases the spread between i and i* (the term in parentheses is persistently positive) for two years. This means that there is persistent profit opportunity and no arbitrage taking in international capital markets. (does this relate to the carry-trade phenomenon?).



Exchange-rate puzzle: Theory predicts the higher the return on foreign investment, the higher should be the demand for the currency, which should appreciate and, therefore, the dollar should depreciate.

VAR results show the opposite: A restrictive foreign monetary policy (i* shock) leads to an appreciation of the US$.


A. Closed Economy models:


I. SR Model Based Identification

Bernanke and Mihov (1998)
They examine the controversies around the liquidity effect (LE) and LR neutrality of M (LRN), simultaneously using an SVAR.

Finding: problems occur when innovations to MP are identified with innovations to monetary aggregates. The existence of LE depends on the identification scheme employed.


Consider the model with

Y=vector of macroeconomic variables (real GDP, GDP price deflator, PGDP, commodity prices, PCOM, and real MS, M2/PGDP)

and

P=MP variables (total reserves, TR, nonborrowed reserves, NBR, Fed fund rate, FFR).


where


The vector of policy indicators have information about policy but also affected by macro variables: P depends on current and lagged values of Y and P and disturbances, one of them being a MS shock . Y depends on its current and lagged values and on lagged values of P. Note that there is a block exogeneity assumption, in the sense that P does not enter Y during the same period, while Y (and P) does enter P during the period. This means that innovations to policy variables do not feedback to the economy contemporaneously. The and are the unobservable, structural shocks, which we want to retrieve. In particular , representing TR demand shock, disturbance to the borrowing function and shock to the stance of MP, respectively.
We are looking for a way to measure the dynamic responses of variables to .

Write the system as a UVAR. Then the policy innovation from UVAR can be written:



or without subscript and superscripts:

u=Gu+Av
This is in SVAR system that relates the observed VAR-based residuals u to unobserved structural shocks v, including. We can identify the system and recover the structural shocks, in particular MS shock, using the impulse response functions.
For identification, B&M consider a model of the market for commercial bank reserves and Fed’s operating procedures expressed in innovation form:
(8)

Commercial banks’ total demand for reserves: innovation in the demand for total reserves (TR) falls with innovation in the Fed fund rate (FF) and rises with a demand disturbance.


(9)

Portion of reserves banks choose to borrow at the discount window: Innovation in the demand for BR rises with innovations in the Fed fund rate and a disturbance term.


(10)

Behavior of FRB: The Fed sets the innovation to the supply of NBR. In doing so, it observes the structural innovations to the demand for TR and BR and responds to structural shocks to TR, BR and MS.


Moreover, . We want to identify the MS shock.
Rewrite the system after substituting for BR:

We can invert the relation
(check)
and get:
# knowns = (n2+n)/2=12/2=6 (estimated from covariances of u’s, the policy block UVAR residuals)

# unkowns = 4 parameters + 3 structural variances = 7.

The model is underidentified.

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