Time series tests of the small open economy model of the current account Birmingham MSc International Macro Autumn 2015 Tony Yates

Intro Note: not examined directly in the MCQ test. Material you can draw on in the main end of year exam. Exploits some of the multivariate time series we covered when we did Clarida-Gali’s work on the DMF model. Reinforces importance of taking macro to the data, and connection between time series econometrics and macro.

Testing we already did: 1 We experimented with stationary and non- stationary endowment processes… … And saw that this flips the sign of the relationship between the ca and output. In the data, current account worsens in a boom. Need non-stationary endowment to get this.

Testing we already did: 2 We derived analytical expressions for the variance of consumption and output growth in terms of the variance of the shock to the endowment. And we got conditions for when var(C)>var(Y) And we noted Uribe’s comment that var(C)>var(Y) in the data for small open economies.

Test 1: ca vs VAR-based infinite stream forecasts First test based on comparison with the SOE’s prediction that the ca=f(stream of future forecasts of endowment growth), and the infinite stream of forecasts using a VAR.

Test 2: unforecastability of ca

SOE TIME SERIES TEST 1: COMPARING MODEL AND DATA BASED INFINITE STREAM FORECASTS

ca as a f(forecast endowments) Recall the definition of the current account. And a relationship we found for consumption in terms of forecast endowments. Now substitute in definition of the current account.

Recall definition of the (eg quarterly) change in a generic time series x_t Derive a relationship between the current account and expected future growth rates of output. We are going to exploit fact that we can also use a VAR to do this.

A VAR fore output and the current account. The VAR’s infinite sequence forecast. Model: ca=infinite sequence forecast. Here, we sub in the VAR into RHS of model eq.

Where does that formula come from?! Picks out output, selects out ca. Matrix formula for the infinite sequence sum, seen many times so far…. And a reminder of what that looks like, expanded….

Where does that come from…ctd! This is a common factor in this infinite sequence, so we can take it out…. Left with a sequence whose first term is this, and where each successive term is also multiplied by this…. This is a reminder of the matrix formula for an infinite geometric sequence, provided abs(max(eig(A)))<1.

This is a definition. This is true, trivially. Ie the current account = itself. For both the top equation on this slide, and the trivial eq to be true, this must also be true, when we use the estimated VAR based forecasts using D_hat.

Results Like all the models in the course…. This one also strongly rejected by the data in Nason and Rogers’ paper.

SOE TIME SERIES TEST 2: ORTHOGONALITY RESTRICTIONS

Recall this expression from our small open economy microfounded model. We can show this to be true. [Try it; use RHS, and expected value of RHS lead one period…] We can also construct these variables from a bivariate VAR representation of the data.

G uses VAR-based terms to replicate the LHS of the top equation. Since top equation=0 [it is a sum of 2 epectations remember]… we should find G=0, or at least is unrelated to current and past values of x_t. Test fails miserably.

Attempting a rescue of the SOE model SOE model assumed only 1 shock, a shock to the exogenous endowment of our SOE consumers. We’ll modify it to include a demand shock [will also give you a clue as to how generally RBC/DSGE models are modified in this way] Leads to a new ‘orthogonality’ or zero condition.

SOE with demand shock We include a serially uncorrelated ‘bliss point shock’ in the SOE model. This wrecks our orthogonality condition. LHS no longer equal to zero.

Explaining why our orthogonality condition is wrecked by that RHS u_t term This is the new Euler equation with the demand shock. [Confirm in an exercise] Recall that this relation holds from the infinite period budget constraint. It turns out by use of the EE, and the law of iterated expectations, that this modified substitution can be made to eliminate the sequence of forecast c’s on the RHS of the infinite period budget constraint.

Deriving our wrecked orthoganality condition We get this after eliminating the sum of infinite future forecast c’s. And then this after a bit of re-arranging, and substituting in defn of the current account.

Using this expression for the current account… Substitute in here to confirm that this equation holds, and therefore expresses the new, wrecked orthogonality condition. We will rescue this shortly to form a new condition and a closely related regression test, that will fail anyway.

Rescuing our orthogonality condition Our old condition, now wrecked by the demand shock. But take expectations conditional on time t-1 of both sides…..and because mu_t is serially uncorrelated, RHS now=0. So we have a new orthogonality condition that leads to similar regression test of our SOE model.

New ‘lagged’ version of our regression test New orth. Condition. VAR-based forecasts conditional on time t- 1 substituted in. D^2 because 2 period- ahead forecasts. Analogous to previous test: we now run regression of G on variables [eg x] dated t-1 and earlier. If model is true, expect zero coefficients. But fails spectacularly again.

Recap We wanted to test implications of the SOE model. Up to this point, we had tests… – … based on the sign of the correlation between the current account and output growth – …and on relative variance of consumption and output growth

In particular We saw that model matches data, which has that current account worsens in good times, only if we assume non-stationary endowment process. And we derived certain plausible restrictions on model parameters that could produce fact from data that var(C)>var(Y) in the SOE.

New tests 2 tests. First test based on equating Second test derives an ‘orthogonality condition’ or really an expression involving actual and expected current accounts that should be zero, therefore uncorrelated with things dated t.

2 nd test modified We saw how enriching the model with a demand shock wrecks the orthogonality conditions. But how we could rescue it by taking expectations of the ‘wrecked’ conditoin at time t-1. Nevertheless, all tests failed.

Remarks Demand shock seems like an ok complication, but many people question what this means [though neuroeconomists believe in them]. Is the demand shock a shock or just a residual indicating model failure? Tests assume rational expectations. All use VAR-based substitutions in place of the infinite stream forecasts.