12/08/2014

Traditional treasury bills and TIPS

TIPS----Treasury Inflation-Protected Securities.

As its name implies, TIPS' major property is that the principal is adjusted to nation's price level, usually measure as CPI. For example, suppose the coupon rates of a ten-year TIPS and a ten-year traditional treasury bill are both 3 percent. If investors buy them at par and hold to maturity, and average inflation is 2.5 for the next 10 years, then the real return of the traditional treasury bill is only 0.5 percent while that of the TIPS is still 3.5 percent. (WHY? Because the principal coupon rate is automatically adjusted to be 2.5+3=5.5 percent) So TIPS keeps the purchasing power.

Ideally, the return yield between TIPS and traditional treasury bill is the average market participants' inflation expectation, which can give us valuable information in inflation forecasting. In real life, however, some other factors also play roles in this yield spread.

(1) Inflation risk:
TIPS is inflation-risk free, while traditional treasury bill is subject to inflation effect, so investors bear more risk buying the latter bond. Based on economic theory, investors should be compensated by some amount to buy the bond, and we call this amount of pecuniary compensation as Inflation Risk Premium.

(2) Liquidity risk:
Based on finance theory, value of a certain type of asset is related to its liquidity. If the asset takes significant time and effort to sell at fair price, then investors might have to lower price to sell it when time constraint is tight. Traditional treasury bills are supposed to be the most liquid assets in financial market, and thus investors of these bonds don't have such worries. TIPS are relatively more illiquid and thus liquidity premium might exist to attract investors to buy TIPS. (TIPS have higher liquidity risk)

To summarize, the yield spread can be expressed as:

y^(n) - y^(r) = inflation expectation + inflation risk premium - liquidity risk premium,

where  y^(n) denotes nominal return of traditional treasury bills, and y^(r) denotes real return of TIPS.

From this equation it is clear that the spread conveys accurate info about inflation expectation only when inflation risk premium is of the same size as liquidity risk premium. However, if both premium are significant smaller than inflation expectation, then the spread still has a good approximation of inflation expectation. Furthermore, if premiums are roughly constant over time, then we can know change of inflation expectation by tracking change of spread rate.






12/06/2014

Paper review of Stock and Watson's Forecasting Inflation

The paper I review is John Stock and Mark Watson's \textit{Forecasting Inflation}, which was published in 1999. The main focus of the paper is on Phillips curve's forecast ability, and authors aimed to answer the four following questions. Has the curve been stable over time? Has the unemployment rate-based Phillips curve done a good job predicting U.S. inflation at 12-month horizon? Is it possible that the curve can be improved by incorporating some other real economic activity variables? If not, does there exist any other model that we can resort to? 
 
First introduced by British economist William Phillips in 1958, the Phillips curve shows that rising price level helps decreasing unemployment rate. However, this inverse relationship doesn't hold in the long run because market participants tend to adjust their inflation expectation and thus as time goes on price level increases while unemployment rate returns back to its natural level. (Friedman, 1968) So in terms of forecasting inflation, researchers typically limit prediction horizons to 12-month so that the inverse relationship exists.
 
In their paper, Stock and Watson (abbreviated as S\&W in the rest of the paper) broadly interpreted Phillips curve as short-run relationship between future inflation and current real market activity. In other words, they aggregate variables in addition to unemployment rate in regressions and examined if they help improve forecast performance of the traditional Phillips curve. To discover existence of better alternative prediction tools, S\&W also considered models either backed by economic theories or more sophisticated in model construction. S\&W used U.S. monthly data from 1959 January to 1997 September on inflation as measured by CPI and Personal Consumption Expenditure (PCE) deflator. The data source is DRI-McGraw Hill Basic Economics Database.
 
The base model S\&W used is as follows: $$\pi^{h}_{t+h}-\pi_{t}=\phi+\beta(L)u_{t}+\gamma(L)\Delta\pi_{t}+e_{t+h}$$ where 
 
\begin{enumerate}
\item $\pi^{h}_{t+h}=\frac{1200}{h}\ln (\frac{P_{t+h}}{P_{t}})$: h-period ahead inflation in the price level $P_{t+h}$, reported at an annual rate
\item $\pi_{t}=1200 \ln(\frac{P_{t}}{P_{t-1}})$: monthly inflation at annual rate
\item $u_{t}$: unemployment rate
\item $\Delta\pi_{t}$: first order difference of inflation
\item $\beta(L)$, $\gamma(L)$: polynomials in the lag operator L 
\end{enumerate}
 
There are two assumptions in the model: inflation data has a unit root and the natural rate of unemployment (NAIRU) is constant.\footnote{Indeed, the natural rate $\bar{u}$ is captured by the constant term $\phi$}. These two assumptions are common in related literature but if future researches find strong evidence against them, then the validity of S\&W's tests are questionable.
 
S\&W first tested whether coefficients in the model have been stable over the sample period. They used Quandt likelihood ratio (QLR) test to discover unknown breakdate. The null hypothesis of QLR(all) is that all coefficients are jointly stable; QLR($\phi,\beta$) tests whether $\phi$ and $\beta(L)$ are jointly stable, assuming constancy of $\gamma(L)$, and QLR($\gamma$) is just the opposite. Table.1 shows their testing result. QLR($\gamma$)s have small p-values, implying that this variable is unstable, and the structural break is 1983. However, QLR test may not be the best testing method, and there could exist more than one structural break. Even though they found model was unstable, S\&W ignored it due to quantitative insignificance of $\gamma(L)$ and regarded it as the benchmark model. 
 
S\&W broadly interpreted Phillips curve as short run relationship between future inflation and current real market activities. The generalized model is as follows: $$\pi^{h}_{t+h}-\pi_{t}=\phi+\beta(L)x_{t}+\gamma(L)\Delta\pi_{t}+e_{t+h},$$ where $x_{t}$ denotes real market variables other than unemployment rate and is either processed or assumed to be stationary. Their detrending method is one-sided Hodrick-Prescott (HP) filter. 
 
 They first used recursive method to estimate in-sample model coefficients and then conducted pseudo out-of-sample forecast. To compare these models' forecast performance, authors used relative mean square error (Rel.MSE) with respect to benchmark Phillips curve and forecast combination regression method. Namely, they regressed the following model: $$\pi^{h}_{t+h}-\pi_{t}=\lambda f^{X}_{t}+(1-\lambda)f^{U}_{t}+\epsilon_{t+h},$$ where 
\begin{enumerate}
\item $f^{X}_{t}$ denotes forecasts based on candidate variable X, made at time t
\item $f^{U}_{t}$ denotes forecasts based on unemployment rate, made at time t
\item $\lambda$ shows how much forecast based on x helps improve forecast ability of benchmark unemployment forecast. 
\end{enumerate}
 
If Rel.MSE is statistically less than 1, then it means its corresponding forecast is better than the benchmark. If $\lambda$ is statistically positive, then the corresponding variable adds forecast accuracy to the benchmark based on unemployment rate.
 
Table.2 and Table.3 tabulated their results. We can observe that in general PCE inflation forecasts are most accurate than CPI inflation ones. Forecast results in the second-half period (1984-1996) are more accurate than the first half. What's more, over years the variables \textit{capacity utilizations} (i.e.IPXMCA) and \textit{manufacturing and trades sales} (i.e.MSMTQ) both have small Rel.MSEs and positive $\lambda$s. So these two variables  help improve forecast that is based on unemployment rate. 
 
To discover potential existence of better models, S\&W also considered models backed by term structure expectations, nominal money supply theory, and multivariate models incorporated with different kinds of variables.\footnote{S\&W used the same methods of coefficient estimation and forecast performance comparison. Since the result tables are too large, I choose not to include them in the paper.} They found that: (1) bivariate models based on term structure of interest rates or nominal money supply produced worse forecast results than Phillips curve; (2) bivariate models incorporating exchange rate or different price indexes didn't outperform Phillips curve consistently; (3) multivariate regressions showed that incorporation of other real activity variables didn't significantly improve the forecasting results. 
 
Based on their findings, S\&W concluded that Phillips curve produced the most reliable short-run inflation forecast among all models they considered and certain aggregate activities variables, namely capacity utilization and manufacturing and trade sales help improve the curve's forecast ability. Their conclusions echoed with some previous researches. For example, Alan Blinder (2001), the former vice chairman of the Board of Governors of the Federal Reserve System, claimed that the curve has done amazingly well over years and it deserves a prominent place in models used for policy-making purposes.  
 
However, there are certain drawbacks in this paper. First of all, since the authors used a large number of forecasts, overfitting bias is inevitable. In addition, they assumed that inflation is I(1), which is controversial. If future researches evidently prove that inflation is integrated of other orders, then this paper's findings might not be valid. Furthermore, S\&W only considered linear models for the sake of simplicity, but as they admitted, the relationship between inflation and real economy variables can be much more complicated. Indeed, Ang, Bekaert and Wei (2006) argued that this paper failed to consider non-linear models like no-arbitrage term structure models or various survey data and thus their findings are incomplete and unconvincing. The harshest critique, however, comes from Atkeson and Ohanian (2001). They used the following random walk model as the benchmark  $$\pi^{12}_{t+12}=\pi^{12}_{t}+v^{12}_{t+12},$$ where $\pi^{12}_{t+12}$ is the forecast of the 12-month rate of inflation and $v^{12}_{t+12}$ is white noise, and used the same forecasting comparison methods as S\&W did. Surprisingly, they found that from 1985 to 2000, none of S\&W's models performed better than this naive random-walk model. This suggests that the best indicator of future inflation is current price level, which, if true, makes researchers' models based on sophisticated theories pretty futile. However, as S\&W commented in their later paper (2008), the robustness of Atkeson-Ohanian model depends delicately on sample periods and forecast horizon. 
 
Forecasting inflation is hard and Phillips curve is merely one candidate model. Recently some researchers (Spen and Corning (2001), Calstrom and Fuerst (2004)) start to use Treasury Inflation-Protected Securities (TIPS) for price level prediction. As its name implies, TIPS are the inflation-indexed bonds whose principal is adjusted to the CPI. If CPI changes, the principal adjusts correspondingly to maintain the same purchasing power. By studying behavior of TIPS, we can gain information on investors' expectation of inflation and do forecast based on that.
 
References
\item Friedman, Milton, "The Role of Monetary Policy", American Economic Review, March 1968, pp 1-17
 
\item Blinder, Alan, "Is There A Core of Practical Macroeconomics That We Should Believe?", American Economic Review, May 1997, pp 240-243.
 
\item Ang, Andrew, Green Bekaert and Min Wei, "Do Macro Variables, Asset Markets, or Surveys Forecast Inflation Better?" Journal of Monetary Economics, May 2007, pp. 1163-1212
 
\item Atkeson, Andrew and Lee Ohanian, "Are Phillips Curve Useful for Forecasting Inflation?", Quarterly Review, The Federal Reserve Bank of Minneapolis, 2001
 
\item Stock, John and Mark Watson, "Phillips Curves Inflation Forecasts", NBER Working Paper 14322, September 2008
 
\item Spen, Pu and Jonathan Corning, "Can TIPS Help Identify Long-term Inflation Expectations?", Federal Reserve Bank of Kansas, Economic Review, Fourth Quarter, 2001
 
\item Carlstrom, Charles and Timothy Fuerst, "Expected Inflation and TIPS", Federal Reserve Bank of Cleveland, November 2004

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