# GG.ComT: Permanent-Transitory Decomposition In CommonTrend: Extract and plot common trends from a cointegration system. Calculate P-value for Johansen Statistics.

## Description

Extract Common Trend(s) from a cointegration system according to Gonzalo and Grange(1995). This method is also known as the Permanent-Transitory Decomposition. Loading Matrix and Othogonal Complement of α and β are also reported.

## Usage

 1 GG.ComT(data, rank, k) 

## Arguments

 data Data used to construct the cointegration system rank Number of cointegration vectors specified k Lag order in VECM

## Details

Currently functions GG.ComT and Kasa.ComT assume that no determinstic parts, such as the constant and the trend, are in the Error-Correction Terms (ECT). So that means we have to keep ecdet = "none" in the ca.jo function (ca.jo is the major function in package urca to estimate cointegration relations). But it does allow the existence of constant term in the VECM (outside ECT).

The method proposed by Gonzalo and Granger decomposes the time series X_{t} as

X_{t}=α(β^{\prime}α)β^{\prime}X_{t}+β_{\perp}(α_{\perp}^{\prime}β_{\perp})^{-1}α_{\perp}X_{t}

where α(β^{\prime}α)β^{\prime}X_{t} is I(0) and the transitory part, and β_{\perp}(α_{\perp}^{\prime}β_{\perp})^{-1}α_{\perp}X_{t} is I(1) and the permanent part (see Equation 11 in Gonzalo and Granger 1995). Be cafreful in Gonzalo and Granger's paper they use different notation for α and β.

Kasa's method decomposes the time series as

X_{t}=β(β^{\prime}β)^{-1}β^{\prime}X_{t}+β_{\perp}(β_{\perp}^{\prime}β_{\perp})^{-1}β_{\perp}X_{t}

where “the first part defines the stationary component and the second part then defines the common stochastic trend” (Kasa 1992) (see Equation 12 in Kasa 1992).

## Value

An object of class ComT.

Fan Yang

## References

Kasa, K., 1992. Common stochastic trends in international stock markets, Journal of Monetary Economics 29, 95-124.

Gonzalo, J., and C. Granger, 1995. Estimation of Common Long-Memory Components in Cointegrated Systems, Journal of Business & Economic Statistics 13, 27-35.

ComT-class,plotComT

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  data(benchmark) x=seq(1,6689,by=23) ## monthly data global=data.frame(benchmark[x,2:4]) GG.ComT (global,2,4) ## Plot the Common Trend G=GG.ComT (global,2,4) Date=benchmark[x,1] plotComT(G,1,x.axis=Date,approx.ticks=12, legend=c("S&P 500 Price index", "Common Trend"), main="Extract Common Trend(s) from Benchmark Markets", ylab="Price", xlab="Time" ) 

### Example output

According to Gonzalo and Grange(1995)

Othogonal Complement of Beta:
[1] 0.6292354 0.4228626 0.6521120

Othogonal Complement of Alpha:
[1] -0.94531972  0.32553268 -0.01997754