SHARP PARAMETER BREAKS 125 Let

D_t =

( 0, t = 1, ..., T^∗ D_t= 1, t = T^∗+ 1, ...T Then we can write the model as:

y_t= (β₁¹+ (β₁²− β₁¹)D_t) + (β₂¹+ (β₂²− β₂¹)D_t)x_t+ ε_t We simply run:

y_t→ c, D_t, x_t, D_t× x_t

The regression can be used both to test for structural change, and to accom-modate it if present. It represents yet another use of dummies. The no-break null corresponds to the joint hypothesis of zero coefficients on D_tand D_t× x_t, for which the “F ” statistic is distributed χ² asymptotically (and F in finite samples under normality).

The General Case

Under the no-break null, the so-called Chow breakpoint test statistic,

Chow = (e⁰e − (e⁰₁e1+ e⁰₂e2))/K (e⁰₁e₁+ e⁰₂e₂)/(T − 2K),

is distributed F in finite samples (under normality) and χ² asymptotically.

8.2.2 Endogenously-Selected Breaks

Thus far we have (unrealistically) assumed that the potential break date is known. In practice, of course, potential break dates are unknown and are identified by “peeking” at the data. We can capture this phenomenon in stylized fashion by imagining splitting the sample sequentially at each possible break date, and picking the split at which the Chow breakpoint test statistic is maximized. Implicitly, that’s what people often do in practice, even if they don’t always realize or admit it.

The distribution of such a test statistic is of course not χ² asymptotically,

126 CHAPTER 8. STRUCTURAL CHANGE let alone F in finite samples, as for the traditional Chow breakpoint test statistic. Rather, the distribution is that of the maximum of many draws of such Chow test statistics, which will be centered far to the right of the distribution of any single draw.

The test statistic is

M axChow = max

τ1≤τ ≤τ2

Chow(τ ),

where τ denotes sample fraction (typically we take τ₁ = .15 and τ₂ = .85).

The distribution of M axChow has been tabulated.

8.3 Recursive Regression and Recursive Resid-ual Analysis

Relationships often vary over time; sometimes parameters evolve slowly, and sometimes they break sharply. If a model displays such instability, it’s not likely to produce good forecasts, so it’s important that we have tools that help us to diagnose the instability. Recursive estimation procedures allow us to assess and track time-varying parameters and are therefore useful in the construction and evaluation of a variety of models.

First we introduce the idea of recursive parameter estimation. We work with the standard linear regression model,

y_t=

t = 1, ..., T , and we estimate it using least squares. Instead of immediately using all the data to estimate the model, however, we begin with a small sub-set. If the model contains K parameters, begin with the first K observations and estimate the model. Then we estimate it using the first K + 1 observa-tions, and so on, until the sample is exhausted. At the end we have a set

8.3. RECURSIVE REGRESSION AND RECURSIVE RESIDUAL ANALYSIS127 of recursive parameter estimates, ˆβ_k,t, for k = 1, ..., K and t = K, ..., T . It often pays to compute and examine recursive estimates, because they convey important information about parameter stability – they show how the esti-mated parameters move as more and more observations are accumulated. It’s often informative to plot the recursive estimates, to help answer the obvious questions of interest. Do the coefficient estimates stabilize as the sample size grows? Or do they wander around, or drift in a particular direction, or break sharply at one or more points?

Now let’s introduce the recursive residuals. At each t, t = K, ..., T − 1, we can compute a 1-step-ahead forecast,

ˆ

The corresponding forecast errors, or recursive residuals, are ˆ

e_t+1,t= y_t+1− ˆy_t+1,t.

The variance of the recursive residuals changes as the sample size grows, be-cause under the maintained assumptions the model parameters are estimated more precisely as the sample size grows. Specifically, ˆe_t+1,t∼ N (0, σ²r_t), where r_t> 1 for all t and r_t is a somewhat complicated function of the data.¹

As with recursive parameter estimates, recursive residuals can reveal pa-rameter instability. Often we’ll examine a plot of the recursive residuals and estimated two standard error bands (±2ˆσ√

r_t).² This has an immediate forecasting interpretation and is sometimes called a sequence of 1-step fore-cast tests – we make recursive 1-step-ahead 95% interval forefore-casts and then

1Derivation of a formula for rt is beyond the scope of this book. Ordinarily we’d ig-nore the inflation of var(ˆet+1,t) due to parameter estimation, which vanishes with sample size so that rt → 1, and simply use the large-sample approximation ˆet+1,t ∼ N (0, σ²).

Presently, however, we’re estimating the regression recursively, so the initial regressions will always be performed on very small samples, thereby rendering large-sample approximations unpalatable.

2σ is just the usual standard error of the regression, estimated from the full sample ofˆ data.

128 CHAPTER 8. STRUCTURAL CHANGE check where the subsequent realizations fall. If many of them fall outside the intervals, one or more parameters may be unstable, and the locations of the violations of the interval forecasts give some indication as to the nature of the instability.

Sometimes it’s helpful to consider the standardized recursive residuals, w_t+1,t≡ ˆe_t+1,t

σ√ r_t,

t = K, ..., T − 1. Under the maintained assumptions, wt+1,t ∼ iidN (0, 1).

If any of the maintained model assumptions are violated, the standardized recursive residuals will fail to be iid normal, so we can learn about various model inadequacies by examining them. The cumulative sum (“CUSUM”) of the standardized recursive residuals is particularly useful in assessing pa-rameter stability. Because

is just a sum of iid N (0, 1) random variables.³ Probability bounds for CU SU M have been tabulated, and we often examine time series plots of CU SU M and its 95% probability bounds, which grow linearly and are centered at zero.⁴ If the CU SU M violates the bounds at any point, there is evidence of parameter instability. Such an analysis is called a CU SU M analysis.

3Sums of zero-mean iid random variables are very important. In fact, they’re so im-portant that they have their own name, random walks. We’ll study them in detail in AppendixB.

4To make the standardized recursive residuals, and hence the CU SU M statistic, opera-tional, we replace σ with ˆσ.