Forecasting Inflation--Surveys Versus Other Forecasts.
Few forecasts are as common in business as forecasts for inflation. Surveys, time-series methods and structural econometric models are the most frequently used means for making such forecasts. In addition, it may be possible to infer future inflation forecasts from the behavior of financial markets. This study takes representative examples from each genre and makes a systematic comparison of how well they performed in the 1980s and 1990s. The results indicate that there is little difference between structural models and surveys. Both are markedly superior, however, to an ARIMA time series model and to inferences from financial markets. Thus, cost and other factors may be the major determinants in each firm's choice between structural models and surveys.
Inflation forecasts play a key role in numerous important business decisions. Management's forecast of inflation, in conjunction with the existing rate of interest, determines the expected real rate of interest. This expected or ex-ante real interest rate, in turn, influences decisions concerning business investment in plant, equipment, technology and inventories. Inflation expectations along with productivity growth are crucial in determining the magnitude of wage and benefit hikes that firms may prudently grant workers. Inflation expectations pertaining to the United States and foreign nations influence business decisions regarding the investment of financial capital at home and abroad. Such decisions impinge significantly on foreign exchange rates thereby triggering changes in export and import activity. Moreover, myriad household decisions are influenced by expected real interest rates, which hinge on the outlook for inflation. Mistakes in forecasting inflation can be quite costly to firms and individuals alike.
There exist several methods of drawing inferences on the magnitude of expected inflation. Several surveys of expected inflation are available today. Econometric models, both of the structural and a-theoretical time-series genres, are also widely used to provide forecasts of inflation. In addition, implicit forecasts of inflation can be derived from financial market phenomena. In this paper, we examine several types of inflation forecasts formulated in the last two decades of the twentieth century and subject them to tests for accuracy and unbiasedness. We evaluate one-year-ahead consumer price index (CPI) inflation forecasts formulated from the first quarter of 1980 through the fourth quarter of 1998 and use actual inflation experienced through the end of 1999 to evaluate the forecasts.
More specifically, we evaluate two major surveys of inflation expectations: the Livingston Survey of professional economists and the Michigan Survey of households.  Both an a-theoretical ARIMA model and a structural econometric model will provide benchmarks for evaluating the forecasting acumen of survey participants. Finally, we will examine inflation forecasts derived from two alternative formulations of Irving Fisher's well-known interest rate framework. 
Evaluating Two Survey Forecasts
Today, there are several surveys of inflation expectations available. Two of the most easily accessible and longest-standing surveys are the Livingston Survey of professional economists and the Michigan Survey of households. In this paper, we will examine these two forecasts.
Joseph Livingston, a Philadelphia financial journalist, initiated the Livingston forecasts in 1946. Livingston queried a sample of professional economists semiannually about their anticipations of the level of several key variables, including the CPI, approximately eight and fourteen months ahead. From such forecasts of the CPI, implicit inflation forecasts can be derived. In the June and December surveys conducted in the 1980s and 1990s, the number of economists in Livingston's sample varied from thirty-seven to sixty-three. The Federal Reserve Bank of Philadelphia now conducts the Livingston Surveys, and the data are available at its website, http://www.phil.frb.org/econ/liv/index.html.  The mean and median one-year-ahead inflation forecasts from Livingston survey respondents, along with the actual inflation rate that ensued, are shown in Figure 1.
Three aspects illustrated in Figure 1 are noteworthy. First, the mean and median Livingston forecasts are extremely close together throughout the ninteen-year interval, typically differing by about one tenth of a percentage point. This may be attributable in part to the economists using similar methodologies in arriving at their forecasts. It may also be due in part to a propensity among economists to fudge their forecasts toward the consensus forecast. This would serve to reduce the likelihood of offering an incorrect forecast that ends up at considerable variance from forecasts of peers, thus potentially leading to a defection of clients. By reducing the frequency and magnitude of outlier forecasts, this propensity may pull the mean forecast closer to the median. At any rate, it is noteworthy that the forecasts of individual forecasters in the Livingston Survey cluster very closely together. Secondly, note the tendency for the forecasts in this period to exceed actual inflation, on average. The last two de cades of the twentieth century was a period of downward-trending inflation, and consensus forecasts tended to lag behind this trend.  This suggests that a fairly strong backward-looking element is present. That is, survey respondents are heavily influenced by recent past inflation. Finally, turning points in inflation expectations typically lag behind turning points in actual inflation. For example, the twelve-month-ahead inflation forecasts rendered in June and December 1998 (last two points plotted) were on a declining path while actual twelve-month-ahead inflation was on a rising trajectory.
The Michigan Survey commenced in 1948. However, the survey was taken sporadically in its early years, and problems inherent in its construction made its interpretation ambiguous. Not until the late 1970s were households asked to provide quantitative, point estimates of expected inflation. The mean and median quarterly forecasts of the more than 500 Michigan respondents, solicited by telephone, are illustrated in Figure 2, along with the actual rate of inflation that subsequently occurred. The Michigan data can be accessed at: http://www.athena .sca.isr.umich.edu.
Note that the Michigan mean inflation forecasts quite consistently and significantly exceed the median forecasts by an average of slightly more than one percentage point--about ten times the corresponding discrepancy for the Livingston Survey. Examination of the individual forecasts indicates that this is attributable to a small proportion of households submitting very high overestimates of inflation. Note that while the mean household forecasts exhibit a strong tendency to overestimate inflation, this tendency is not apparent in the case of the median forecast. As is the case with the Livingston Survey, turning points of Michigan household forecasts typically lag turning points in actual inflation.
In our judgment, the median forecast is superior to the mean as a consensus measure of forecasting because it gives outliers less weight and treats all forecasters equally. The mean essentially gives too much weight to incompetent or uninformed respondents. In our tests to follow, we will therefore use the median forecasts of the Livingston and Michigan surveys.
ARIMA Inflation Results
The ARIMA (autoregressive integrated moving average) procedure has been utilized for more than two decades by professional economists and other analysts for generating inflation and other forecasts. While efficiently incorporating the actual history of inflation into its forecast, the ARIMA method ignores current information that sophisticated agents might utilize in forecasting inflation. For example, in the first half of 2000, two types of supply shocks were simultaneously occurring that would likely influence inflation during the remainder of 2000 and beyond. The sharp increase in oil prices during 1999 and early 2000 constituted an adverse supply shock tending to raise the future inflation rate. On the other hand, unexpectedly strong productivity growth constituted favorable supply shocks tending to inhibit forthcoming inflation. In addition, wealth effects emanating largely from the tremendous appreciation of stock values in the previous five years were producing positive aggregate demand shocks. These wealth effects and the coincident sustained increase in consumer confidence were working to boost inflation.
All of these ongoing developments are ignored in the a-theoretical ARIMA framework, which focuses entirely on past inflation in developing its forecasts. A failure of survey respondents to outperform the forecasts generated by the ARIMA model would therefore suggest a failure of such respondents to factor in the likely influence of ongoing developments relevant to future inflation. The ARIMA forecasts thus serve as a useful benchmark.
Forecasts from a Structural Model of Inflation
A structural inflation model similar to one employed by the Federal Reserve Bank of San Francisco (described in Throop, 1988) was utilized to derive relatively sophisticated forecasts of inflation. Forecasts generated by this model may be used as a rigorous benchmark in evaluating various other inflation forecasts. Our structural model incorporates measures of aggregate demand and supply-side phenomena as well as the behavior of past prices in developing forecasts of inflation. The structural inflation model used in this study is defined in equation 1.
(1) [P.sub.t] = [a.sub.1] GAP + [[[sigma].sup.3].sub.i=1] [b.sub.i][POIL.sub.t=i] + [[[sigma].sup.5].sub.i=1] [c.sub.i][REX.sub.t-i] + [[[sigma].sup.15].sub.i=1] [d.sub.i][P.sub.t=i]
In this model, [P.sub.t] represents the quarterly rate of change of prices (CPI) predicted by the model.  GAP, included to capture the important role of cyclical forces in the inflation process, is a measure of the output gap--the deviation between actual and potential real GDP. This latter measure is obtained by passing the real, chain-weighted GDP series through the Hodrick-Prescott filter.  [POIL.sub.t-i] is the quarterly rate of change of oil prices and is intended to pick up important aggregate supply shocks that impinge on inflation. REX is the quarterly rate of change of real exchange rates measured by the J.P. Morgan trade-weighted index of real exchange rates. Exchange rate changes influence U.S. price level phenomena in at least three ways. First, exchange rate changes directly influence the dollar cost of goods imported into the United States. Second, changes in exchange rates influence aggregate demand for U.S. goods and services by impacting the international trade balance (exports less im ports). Finally, changing exchange rates influence the amount of discipline that U.S. firms must exercise to maintain a competitive cost and price structure vis-a-vis foreign competitors. A real appreciation of the U.S. dollar, by reducing the cost of imports, reducing aggregate demand for U.S. goods and services and increasing the pressure on U.S. firms to hold down prices to remain competitive, tends to reduce the U.S. rate of inflation. Finally, in Equation 1, [P.sub.t-i] represents recent past rates of inflation. This variable is intended to pick up inertia that characterizes the inflation process.
To capture the dynamics of the inflation process, the model allows several lags of past oil prices, exchange rate movements and actual inflation to influence the current inflation rate. We experimented with various lag structures of these variables and found the results to he relatively robust with respect to small changes in the lag structures. After experimenting, we settled on the lag structure that provided the best fit. This involved three quarterly lags on past oil price inflation, quarterly lags on past real exchange rate changes, and fifteen quarterly lags on past rates of change of consumer prices.
The structural model essentially forecasts inflation each quarter. Then, incorporating new information, it re-estimates the regression, and produces a new forecast for the next period. Because the structural inflation model forecast constitutes a rigorous standard, the ability of other forecasts to approximate this forecast in terms of accuracy and other forecasting criteria would be an impressive indication of the quality of such forecasts. The inflation forecasts yielded by our structural model, along with the ARIMA forecasts and the actual inflation rate that prevailed in the past two decades, are shown in Figure 3.
Note that the ARIMA model tends to yield positive forecast errors (overestimates of inflation) when inflation is falling and negative forecast errors when inflation is rising. Also, the turning points in the ARIMA forecasts lag turning points in actual inflation. These characteristics of ARIMA are inevitable, given its total reliance on past inflation in generating forecasts. The structural model provides a relatively tight fit, and turning points yielded by its forecasts often coincide with turning points in the actual rate of inflation. Unlike the survey and ARIMA forecasts, the structural model forecasts did not exhibit upward bias in forecasting the low inflation rates of the late 1990s. As we will note later, the explanatory power of the structural model exceeds that of all other forecasts.
Selecting from Four Fisher Model Forecasts
Irving Fisher's well-known model expressing the relationship between nominal interest rates and expected inflation is given in Equation 2.
(2) I - r = [P.sup.e]
In this formulation i represents the nominal interest rate, r the ex-ante or expected real interest rate, and Pe the expected rate of inflation. Numerous empirical studies have noted the high correlation between expected inflation and the nominal interest rate. A strong version of Fisher's hypothesis is that changes in expected inflation ([P.sup.e]) are reflected in one-for-one changes in nominal interest rates (i). Rearranging the equation to solve for expected inflation, we obtain equation 3.
(3) [P.sup.e] = i-r
We employed the one-year Treasury bill yield to represent i. To derive forecasts for one-year-ahead expected inflation via this framework, one must make some assumption about how agents formulate their expected real rate of interest, r. We experimented with four different assumptions regarding the formulation of r. In the first formulation, we assume agents set their expected real rate equal to the most recent ex-post real interest rate. For example, a financial market participant forecasting inflation in November 1998 (fourth quarter of 1998) is aware of the actual real rate that was realized (ex-post real rate) on a one-year Treasury bill purchased in October 1997. If that real rate actually realized was two percent, and if the Treasury bill yield in November 1998 was five percent, the Fisher one-year-ahead inflation forecast rendered in November 1998 would be three percent.
In the second formulation, we assumed agents establish the expected real rate by taking an average of ex-post real rates over the most recent four quarters. The third formulation used an eight-quarter period average of ex-post real rates to formulate the expected real rate.
Finally, in the fourth Fisher formulation, we assumed agents establish their expected real interest rate at a constant level equal to the average actual real rate earned on one-year Treasury bills that occurred over the entire 1980:1 - 1998:4 period (3.93 percent). This, of course, is a heroic assumption because agents were not aware of the magnitude of this average real rate at the time the forecasts were made. This assumption constrains the magnitude of the average forecasting error over the full period to zero, and allows us to ascertain whether, even under stringent assumptions, the Fisher framework can stack up well against alternative methodologies in forecasting inflation. As we will see, it does not.
The first and fourth formulations of the Fisher framework provided the most accurate inflation forecasts among the four formulations. Figure 4 illustrates these two Fisher inflation forecasts, alongside actual inflation that ensued.
Note via visual inspection of Figure 4 that the Fisher forecasts fail to track actual inflation as well as the forecasts generated by the surveys, by the ARIMA Model and by the structural model (review Figures 1, 2, and 3). This will be documented shortly via formal tests. If expected inflation were the only factor influencing one-year Treasury bill yields, the Fisher formulations would likely track inflation very well. However, many other factors besides the outlook for inflation trigger changes in Treasury bill yields. Liquidity effects engineered by the Federal Reserve clearly influence short-term yields, as do business cycle developments, government budget deficits, safe-haven considerations and other factors. The cyclical amplitude of short-term yields is relatively high, typically exceeding that of long-term yields. This fact is responsible in part for the fact that the amplitude of Fisher forecasts illustrated in Figure 4, unlike that of the survey forecasts, exceeds that of actual inflation.
In the figures, the large positive Fisher forecast errors of the early 1980s can be attributed to the regime of restrictive monetary actions maintained by the Volcker Federal Reserve as it pursued its policy of disinflation.  In addition, the large federal budget deficits in the 1981-1986 period likely exerted upward pressure on yields and thereby contributed to the propensity of the Fisher models to overpredict inflation in that period. The large negative forecast errors of 1992 and 1993, particularly evident in the case of Fisher 4, are the result of the sluggish initial recovery of the U.S. economy from the 1990-1991 recession and the consequent targeting by the Fed of interest rates at very low levels in an effort to jump-start the fledgling economic expansion.
Statistical Analysis of the Six Inflation Forecasts
We subjected the six inflation forecasts to a battery of simple statistical tests to obtain measures of the accuracy and a crude measure of bias in the forecasts. The mean error (ME) statistic simply sums the positive and negative forecasting errors and divides by the number of forecasts (periods). Hence, a mean error of zero indicates that the forecasts are correct "on average". A positive mean error indicates a propensity to overestimate inflation on average, while a negative mean error indicates a tendency to underestimate inflation. The ME may be considered a simple measure of forecasting bias. The average absolute error (AAE) and the root mean square error (RMSE) are measures of forecasting accuracy. As its name suggests, the average absolute error sums up the absolute value of the forecast errors of all periods and divides by the number of periods. The RMSE sums the squares of the forecast errors, divides by the number of periods and takes the square root of the resulting magnitude. Hence, the RMSE mea sure levies a disproportionate penalty to large forecasting errors relative to the AAE measure.
Note that a particular type of forecast may be correct on average (zero mean error) and yet be extremely inaccurate as indicated by very large average absolute errors and root mean square errors. Table 1 provides these simple forecasting statistics for each of the six inflation forecasting methods previously discussed for the full 1980:1-1998:4 period, and for the 1980s and 1990s taken separately.
Looking first at the mean error criterion, the data indicate that both the Livingston economists and the ARIMA model significantly overestimated inflation on average over the full period. In the ARIMA, this follows from the fact that the framework is strictly backward-looking and that inflation was trending downward through the full period. In the case of the Livingston respondents, it suggests that past inflation weighs rather heavily in the formulation of forecasts by professional economists. The median Michigan household forecast exhibits a negligible mean error over the full period, as did Fisher 4 (via construction). The structural model and Fisher 1 overestimated inflation on average by about a third of a percentage point.
Looking at the 1980s and 1990s separately, it is clear that economists were consistent in their tendency to overestimate inflation. The Michigan households tended to underestimate inflation in the 1980s and overestimate it in the 1990s, thus explaining the negligible mean error for the full period. As is intuitively plausible, the ARIMA mean errors were larger in the 1980s than the 1990s because the rate of descent of inflation was greater in the former decade. Largely due to more restrictive monetary policy and larger federal budget deficits in the 1980s than the 1990s, the Fisher forecasts yield positive forecast errors in the 1980s and negative errors in the 1990s. Finally, the structural model does a better job in the 1990s than in the 1980s.
In terms of accuracy (AAE and RMSE), note that the performance of the Livingston and Michigan median forecasts are similar to that of the structural model. While the structural model barely noses out the surveys on the AAE criterion, both surveys narrowly defeat the structural model on the RMSE criterion. In terms of the AAE and RMSE, the ARIMA forecasts are ranked in the middle while the two Fisher forecasts finish last.
Looking at the sub-periods, all forecasts improve in the 1990s era of lower inflation, and the rankings remain pretty much intact. It is interesting to note that in this recent period both survey forecasts edge out the structural forecasts in terms of both AAE and RMSE, although the differences are not statistically significant. In the 1980s, the results of comparisons between the surveys and the structural model are mixed. In general, the survey forecasts are more accurate than ARIMA and Fisher forecasts and are approximately on a par with the structural model forecasts.
Testing the Six Forecasts for Unbiasedness
Scatter plots showing the actual inflation rates and the forecasts produced by the Livingston median, Michigan median, and structural model are provided in Figures 5, 6, and 7, respectively. In each case, the forecast of inflation is shown on the horizontal axis and the actual rate that ensued is depicted on the vertical axis. The points representing the 1980s are shown in solid circles and the points representing the 1990s are shown in hollow squares. In each diagram, a 45-degree guideline is drawn. Points falling above the guideline indicate that forecasters underestimated inflation (negative forecast errors) while points lying below the line indicate respondents overestimated inflation (committed positive forecast errors).
A formal test of unbiasedness of inflation forecasts can be conducted by estimating a regression of the following form:
(4) P = a + b[P.sup.e]
The null hypothesis of unbiasedness is that the intercept (a) is zero and the slope (b) is one. In other words, if the forecasts are unbiased, a regression line fit through the points depicted in the figures should not differ significantly from the 45-degree line.  While regression lines are not shown in the figures, it appears that regressions fit through the points in Figures 5, 6, and 7 track the 45 degree line fairly well. This is not the case, however, in the ARIMA and Fisher forecasts (figures not shown). The formal regression results are shown in Table 2.
The table reports the estimated intercept and slope for each forecast, with standard errors reported below these coefficients in parentheses. Also reported are [R.sup.2]s--the proportion of the variation in actual inflation that can be accounted for statistically by variation in inflation forecasts. The right-hand column of the table indicates whether the null hypothesis of unbiasedness can be rejected at conventional levels (.05).
Note first in the case of the Livingston and Michigan median inflation forecasts that neither the intercepts nor the slopes differ significantly from their hypothesized values of zero and one, respectively. In both instances, variation in forecasted inflation accounts statistically for more than two thirds of the variation in actual inflation. In neither case can the hypothesis of unbiasedness be rejected. Similar results obtain in the case of the structural model, which possesses the highest explanatory power ([R.sup.2]) among the six forecasts. Unbiasedness cannot be rejected in this model at conventional levels of significance. Note, however, that in the ARIMA and both Fisher forecasts, the estimated intercepts are considerably greater than zero and the estimated slopes considerably lower than 1.0. These differences from the hypothesized values are quite significant statistically, and the hypothesis of unbiasedness is easily rejected in each case. Also, the R2s are much lower; that is, the explanatory pow er of the ARIMA and Fisher forecasts in explaining actual inflation are inferior to that of the two surveys and the structural model.
In Conclusion, Surveys Deserve More Consideration
If not adequately anticipated, inflation can have important adverse consequences for business firms. Firms may therefore expend considerable resources to obtain forecasts of inflation. This study demonstrates, however, that median forecasts of survey respondents may constitute a less expensive and, on average, not appreciably inferior source of information about forthcoming inflation than formal modeling. Median inflation forecasts of the Livingston and Michigan survey respondents are found to be unbiased in the past 20 years, and dominate forecasts of a strictly backward-looking ARIMA model, as well as forecasts generated by the Fisher interest-rate framework. This implies that both economists and households include relevant contemporary information in addition to past inflation in formulating their inflation forecasts.
A structural inflation model that incorporates supply-side forces and business cycle phenomena as well as past inflation exhibits more inflection points in predicting inflation and exhibits slightly superior explanatory power in our tests for unbiasedness. However, the median survey forecasts have been approximately as accurate as the structural model in forecasting inflation in the last two decades of the twentieth century. Because the survey information is virtually costless, a cost-benefit analysis suggests that many firms might accord more weight to surveys in implementing plans to circumvent potential costs associated with mistakes in forecasting inflation.
The authors would like to thank Jae Sung Kang for able research assistance, including the construction of the graphs in this paper.
Lloyd B. Thomas, Jr. is Professor of Economics at Kansas State University. He holds BA and MA degrees in economics from the University of Missouri and a Ph.D. in economics from Northwestern University.
Alan P. Grant is an Assistant Professor of Economics at Eastern Illinois University. He received his Ph.D. from Kansas State University.
(1.) Another well-known and easily accessible survey of expected inflation is the Survey of Professional Forecasters, conducted quarterly by the Federal Reserve Bank of Philadelphia. Because that survey did not commence until the latter part of 1981, we do not evaluate its performance in this paper. The Wall Street Journal also conducts surveys of economists, as does the National Association for Business Economics and the commercially available Blue Chip Economic Indicators.
(2.) A very promising implicit measure of inflation expectations over a particular time horizon can be derived by subtracting the yield on indexed Treasury securities from the yield on unindexed Treasury securities of the same maturity. Unlike many survey respondents, financial market participants have a powerful incentive to formulate accurate and unbiased forecasts of inflation. However, indexed government bonds have only been available in the United States since 1997. Hence, it is too early to evaluate the performance of inflation forecasts derived via this methodology.
(3.) The Survey of Professional Forecasters inflation forecasts are also available at this web site.
(4.) When inflation was ratcheting upward during the 1960-1980 period, forecasts of survey respondents persistently underestimated inflation. On this, see Thomas (1999).
(5.) The rates of change of the variables are measured as the change in the natural logarithm each quarter. The algorithm utilized generated forecasts for the change in log CPI for each of the next four quarters. The changes were added up to get the four-quarter-ahead log CPI, and this was used to calculate the one-year-ahead inflation forecast.
(6.) The Hodrick-Prescott filter allows the econometrician to separate the GDP time series into two components: a flexible (nonlinear) trend and cyclical fluctuations of GDP about that trend. The Hodrick-Prescott filter fits a trend line through the GDP series similar to that which an individual would draw if asked to freehand a smooth curve through the series. For a more technical description, see Robert Hodrick and Edward Prescott (1997).
(7.) In October 1979, in order to implement its battle against inflation more effectively, the Federal Reserve abandoned its policy of setting explicit targets for the federal funds rate. Instead, it set targets for growth of the monetary aggregates (hence, the policy was dubbed "The Monetarist Experiment"). This policy was maintained for approximately three years and during this period the federal funds rate and one-year Treasury bill yield averaged 14.00 percent and 12.75 percent, respectively. In the five-year period from the beginning of 1980 through the end of 1984, these yields averaged 12.26 and 11.35 percent, respectively. These extraordinarily high yields, triggered initially by restrictive Fed actions and later reinforced by the emergence of large federal budget deficits, caused the Fisher model to significantly overestimate inflation in that period.
(8.) Note that this test is superior to the mean error as a measure of forecasting bias. Suppose hypothetically that forecasters always predict five percent inflation in a time interval in which actual inflation ranges from zero to ten percent and averages five percent. While the mean error is zero in this case, the points in the scatter plot would fall along a vertical line intersecting the horizontal axis at five percent expected inflation, In the case, the joint test of zero intercept and unitary slope indicates the forecasts are biased, while the mean error fails to reveal the bias.
Croushore, Dean, "Inflation Forecasts: How Good Are They?" Federal Reserve Bank of Philadelphia Business Review, May/June 1996, pp. 15-25.
Fama, Eugene and Michael R. Gibbons, "A Comparison of Inflation Forecasts," Journal of Monetary Economics, May 1984, pp. 327-48.
Gramlich, Edward M., "Models of Inflation Expectations Formation: A Comparison of Household and Economist Forecasts," Journal of Money, Credit, and Banking, May 1983, pp. 155-73.
Hodrick, Robert J. and Edward Prescott, "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit, and Banking, February 1997, pp. 1-16.
Keen, Howard, "Economists and Their Forecasts: Have the Projections Been That Bad?" Business Economics, January 1987, pp. 37-40.
Laster, David, Paul Bennett, and In Sun Geoum, "Rational Bias in Macroeconomic Forecasts," Quarterly Journal of Economics, February 1999, pp. 293-318.
Thomas, Lloyd B., "Survey Measures of Expected U.S. Inflation," Journal of Economic Perspectives, Fall 1999, pp. 125-44.
Throop, Adrian, "An Evaluation of Alternative Measures of Expected Inflation," Federal Reserve Bank of San Francisco Review, Summer 1988, pp. 27-43.
PERFORMANCE OF ALTERNATIVE INFLATION FORECASTS IN 1980S AND 1990S (ONE YEAR-AHEAD CPI INFLATION FORECASTS) Average Root Mean Absolute Squared Forecast Mean Error Error Error A:FULL PERIOD (1980:1-1998:4) Livingston Median 0.62 0.89 1.16 Michigan Median -0.04 0.89 1.14 ARIMA 0.64 1.31 1.89 Structural Model 0.36 0.88 1.17 Fisher 1 0.30 2.11 2.92 Fisher 4 0.00 1.94 2.53 B:EARLY PERIOD (1980:1-1989:4) Livingston Median 0.73 1.12 1.45 Michigan Median -0.37 1.16 1.40 ARIMA 0.74 1.82 2.42 Structural Model 0.62 1.07 1.42 Fisher 1 0.60 2.89 3.75 Fisher 4 1.11 2.50 3.08 C:LATE PERIOD (1990:1-1998:4) Livingston Median 0.50 0.62 0.70 Michigan Median 0.32 0.59 0.76 ARIMA 0.53 0.75 1.01 Structural Model 0.08 0.67 0.80 Fisher 1 -0.02 1.24 1.54 Fisher 4 -1.23 1.31 1.72 TESTS FOR UNBIASEDNESS OF INFLATION FORECASTS 1980:1-1998:4 P = a + b[P.sup.e] + [e.sub.t] Reject Forecast a b R2 Unbiasedeness  Livingston Median -0.18 0.90 .74 No (0.44)  (0.11) Michigan Median -0.12 1.04 .68 No (0.57) (0.15) ARIMA 1.39 0.53 .59 Yes (0.49) (.011) Structural Model 0.31 0.88 .88 No (0.36) (0.09) Fisher 1 2.27 0.37 .56 Yes (0.42) (0.09) Fisher 4 2.36 0.38 .37 Yes (0.39) (0.15) Notes: (1.)Figures in parentheses reported below estimated coeffients are standard errors. (2.)Reported results are based on chi-square tests of joint null hypothesis (a,b) = (0,1) of equation P = a +b[P.sup.e] + [e.sub.t] using 95 percent confidence level as criterion.