Replication of Vinod's fractional cointegration in JSPI and porting to R.

Lee, Jennifer

I attempted to replicate Vinod's (2002) article on fractional Cointegration by porting it to the freely available R software instead of Gauss. The usual cointegration is not detected and unit root tests have lower power in testing for long run equilibrium. This is because economic variables are instead "fractionally" integrated (FI) of order d. According to Walrasian general equilibrium, there exists some weak relation or tie among all economic variables. The measurement of the strength of tie (SOT) is more important than the statistical testing for the mere existence of a tie. The SOT is an index that quantifies market responsiveness.

The process for estimating tie integration is composed of six steps: (1)Estimate [d.sub.xj] for each time series [x.sub.jt] ~ FI([d.sub.xj]). This d parameter explains the equilibrium or long memory structure. (2)Use linear regression to get the relationship among [x.sub.jt]'s. Estimate the residuals [z.sub.t]. (3)Obtain and compute the [d.sub.z]. (4)Derive a [d.sub.null] occurring in the absence of market forces by minimization of the functional equation. (5) Solve for the strength of tie (SOT). (6) Choose an appropriate value of the SOT to determine whether the evidence for tie integration is strong enough for the context at hand.

The paper uses 2 monthly data sets, one on the stock market dividends and prices, and the second on US wages and prices. I used R language to replicate the paper. R is a freeware and is derived from the S language and environment. The Vinod articles used S Plus for estimation.

First, all of the macroeconomic variables are tested and found to be unit root processes. Therefore, the first difference of the variables, which is found to be stationary, is used. This is used to estimate the fractional integration parameter. In another set of estimation, the first difference of the logarithm of the variable (growth rate) is used.

In order to estimate the fractional cointegration parameter d, the package called fracdiff in R is installed. Fracdiff calculates the maximum likelihood estimators of the parameters of a fractionally-differenced ARIMA (p,d,q) model, together with their estimated covariance and correlation matrices and standard errors, as well as the value of the maximized likelihood. Another method of estimating the cointegration parameter d is through the GPH (Geweke-Porter-Hudak) method. (1)

The next step will be to find the [d.sub.null] This is the solution to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The minimization function can be done in R using the code nlm.

I was not able to fully replicate the paper. The estimates of the d, and hence the strength of tie were different from the article. The SOT in the article was 48 for the stock market and 25 for the labor market. My estimates of the SOT were 43 and 14, respectively. Both conclude that the SOT for the stock market is larger than the SOT for the labor market, suggesting that the stock market absorbs its shocks faster than the labor market.

REFERENCES

Vinod, HD. "A looser cointegration concept using fractional integration parameters and quantification of market responsiveness" Journal of Statistical Planning and Inference 100(2002) pages 399-410.

Maddala, G.S. and Kim, I. Unit Roots, Cointegration and Structural Change. Cambridge University Press: Cambridge 1998.

NOTES

(1.) GPH proposed a semi-nonparametric procedure to estimate d. The d can be estimated from the following least squares regression: ln(I([[bar.[omega].sub.j])) = c - d In(4 [sin.sup.2] ([[bar.[omega].sub.j] / 2)) + [eta], where [[omega].sub.j] is the frequency and I([[omega].sub.j]) is the periodogram of X at that frequency. (Maddala and Kim, 1998).

Jennifer Lee, Fordham University, jlee@fordham.edu.