If we assume that parameters are not constant over time, we can split the entire sample into two or
more sub-samples; each sub-sample corresponds to a specific regime, determined by the estimated
value of a selected threshold variable (th). It is reasonable to assume that agents respond
asymmetrically to shocks provided they expect different scenarios; as well as it is plausible to
analyse differently the response of financial variables to macroeconomic news in good or bad times.
Technically, threshold estimates are obtained by minimizing the sum of squared residuals in both
regimes; computationally the minimization process takes the form of a grid search. Once the
threshold value is determined, the estimation process is equivalent to estimating a regression with
deterministic dummy variables. The threshold effect is simply denoted by the difference of
parameter estimates in sub-regimes.
Hansen has proposed a statistical test to check for the presence of a threshold. Consider the null
hypothesis H0: τ0 = τ, where τ0 is the true value of the threshold, while τ an arbitrarily fixed
value. To test the hypothesis on the threshold we use the following F statistics:
LRn (г) = n S⅛S(г) (7)
S(г)
where n is the number of observations, and S (г) is the concentrated sum of squared residuals. The
likelihood ratio statistics has a non-standard distribution which depends on η2 . In particular, η2 is
equal to unity in case of homoscedasticity; while in case of heteroscedasticity it must be estimated
(Hansen, 2000). Test results are valid asymptotically; so that n needs to be greater than 100. The
sequence of the likelihood ratio is then used to obtain 95% confidence intervals.
Consider now the Campbell-Shiller equation (5). In the top panel of Figure 3 we plot the likelihood
ratio sequence against the threshold variable (tptn,m). The left diagram refers to the pair of maturities
(60, 3); the right diagram show results for the combination (120, 3). In both cases the LR sequence
breaks the 5% critical value line suggesting the presence of a breakpoint, i.e. one threshold. Results
for the other couples of maturities (n, 3) are similar; test results reveal the presence of two regimes.
|
Confdence Internal Construction fcr Threshold | ||
|
Lkeihozod Ratio Sequeice in Gema | ||
|
0'-------------------'-------------------'-----------■—∙—'------------------,-------------------'------------------- -4 -2 0 2 4 6 ThreshOd Variabetpt60V3 | ||
|
1400 1200 I 1000 I 800 S 600 1 400 200 |
Confdence Interval Construction for Threshold | |
|
4 -2 0 2 4 6 8 Threshold Variableztpt603 | ||
Figure 3
12
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