Kim and Ryoo (2011)

In a long-term study done using a data set beginning in the 1900’s and ending in 2009 in the United States, Kim and Ryoo (2011) used a two-regime threshold vector error correction model (TVECM) to test for cointegration. This test used the TVECM to allow for asymmetric adjustments of stock return and expected inflation towards the long-run equilibrium which distinguished the results from those of past studies. The need to incorporate asymmetric

adjustments between stock return and inflation stems from the independent findings of recent studies (Barnes et al., 1999; Kim, 2003) which ultimately provide a compelling argument for the use of the TVECM model as opposed to the Vector Error Correction Model (VECM) as has been commonly used in previous studies of a similar nature

Kim and Ryoo (2011) used the TVECM to test for the condition of unity of the long-run elasticity between the stock price and the goods price in order to determine whether equities can effectively act as a hedge against inflation. The authors note that this condition of unity stems from the Fisher Hypothesis in which a change in the nominal return on stock occurs subsequently to a change in the expected rate of inflation. The study conducted by Kim and Ryoo (2011) used the TVECM in order to allow for asymmetric adjustment of both the stock return and expected inflation variables toward their long-run equilibrium levels in order to address the disparity in the results obtained by past studies which have failed to account for asymmetric adjustment. The directions of the asymmetric adjustments towards the long-run equilibrium were dependent on whether stocks were overpriced or underpriced relative to the price of goods (Kim and Ryoo, 2011).

The model used in this study was based on Seo's (2006) test for cointegration in which the TVECM was used to test a null hypothesis of linear no cointegration against an alternative hypothesis of threshold cointegration. This test was therefore an extension of common tests for cointegration which were typically limited to testing for an alternative hypothesis of linear cointegration. The TVECM model used in the study by Kim and Ryoo (2011) was specified as:

𝛷(𝐿)-𝛥𝑋_𝑡= 𝛼₁𝑍_𝑡−1𝐼(𝑍_𝑡−1≤ 𝛾) + 𝛼₂𝑍_𝑡−1𝐼(𝑍_𝑡−1> 𝛾) + 𝜇 + 𝜀_𝑡 where:

- 𝛷(𝐿) is the lag of polynomial of order p;

- ∆Xt is the first difference of Xt (Vector of the stock return and inflation);

- α1 and α2 are the vectors of speed of adjustment coefficients; I(•) is an indicator function which takes the value 1 if the condition inside the brackets is satisfied

- 𝜀_𝑡 is a vector of i.i.d. error terms

- Zt takes the place of St - Pt, which represents the deviation from the long-run equilibrium while 𝛾 is its long-run mean. (St is the log of the stock price, while Pt is the log of the goods price).

In the above model there are two possible regimes represented by the terms in the brackets. In Regime 1 where (𝑍_𝑡−1≤ 𝛾) stocks are underpriced relative to goods and in Regime 2 where (𝑍_𝑡−1> 𝛾) stocks are overpriced relative to goods.

Kim and Ryoo (2011) conducted the test for a long-run relationship between the stocks and goods price using moving subsample windows, using monthly data between 1900 and 2009 in the United States. The stock price data was sourced from the S&P 500 and the Dow-Jones Industrial (DJI) indices while the Consumer Price Index (CPI) was used as the inflationary variable. Graphs of the S&P 500 stock return and the Consumer Price Index during the sample period are replicated below (Figure 2) in order to provide a comparison of the volatility of the variables in the United States and those published by Alagidede and Panagiotidis (2010) in Figure 2.

The authors make an important note in that they refuse to distinguish between actual and expected inflation in the context of the study and use actual goods prices as a proxy for expected goods price, which they based on the findings of Madsen (2007) who determined that the testing outcomes in such a context can be biased when incorporating model-based expected inflation values. The moving subsample windows of 30 years were used because the authors have found that this period length ensures that the results are stable, which is consistent with the sample size requirement for this test as specified by Seo (2006). The authors used the ADF test when the moving subsample window covered a period of 30 years, and found that in most cases the time series are integrated of order one, showing that should a standard linear regression test be employed that the long-run relationship between the goods price and the stock's price would be spurious.

The results of the study indicated that the null hypothesis of no linear cointegration was in all cases rejected in favour of the alternate hypothesis of threshold cointegration at the 10% level of significance from 1980 onwards. The study found that a parallel relationship existed between the long-run stock price and the goods price and therefore that US stocks acted as an effective hedge against inflation since the 1950's, which lends support to the Fisher Hypothesis. The study also found the presence of asymmetric error corrections of common stock returns and expected inflation.

Figure 2: Time plots of the S&P500 stock return and the CPI inflation rate between 1900 and 2010 in the US.

Source: Kim and Ryoo (2011).