Statistical Models of Stocks and Bonds Zachary D Easterling: Department of Economics The University of Akron Abstract One of the key ideas in monetary economics is that the prices of investments tend to move in the opposite direction as interest rates. Is there a correlation between stocks, bonds and housing and do they move in the same direction? Can an economist predict the movement of one by looking at the aggregate of the others in relation to interest rates? Using Simple and Multiple Linear Regression, Chi- Square, One-Sample, and Two-Sample T-tests, this short work will focus on any positive or negative correlation between the Federal Funds Rate and investment media.
Simple Linear Regression: Using the Federal Reserve Federal Funds Rate to predict the yield to maturity on a One-Year Treasury Security. H 0 : The Federal Funds Rate is not a useful predictor of the One-Year Treasury Security yield. H a : The Federal Funds Rate is a useful predictor of the One-Year Treasury Security yield. Figure 1: MiniTab Output for Simple Linear Regression The regression equation is One Year Security = 1.38 + 0.758 Federal Funds Rate Predictor Coef SE Coef T P Constant 1.3772 0.2483 5.55 0.000 Fed Funds Rate 0.75839 0.05592 13.56 0.000 S = 1.29798 R-Sq = 49.2% R-Sq(adj) = 48.9% MiniTab was used to predict the yield to maturity on a One-Year Treasury Security using the Federal Funds Rate; which is set daily by the Federal Reserve Bank. The output generated by MiniTab in conjunction with the hypotheses made in a simple linear regression leads me to accept the idea that the Federal Funds rate is indeed a useful predictor of the One-Year Treasury Security because of the incredibly small P-value indicated in the output. The R 2 values indicate however, that only 48.9% of the variability in the One Year Treasury is explained by the movement of the Federal Funds Rate. Unfortunately, the residual plot information for the Simple Linear Regression is somewhat suspect. While the Versus Fits scatter plot does indicate a random cloud of residuals, the Histogram does not resemble a clear Normal Analysis of Variance Curve. In this case the Histogram appears more right Source DF SS MS F P Regression 1 309.92 309.92 183.96 0.000 skewed, and may therefore lead Residual Error 190 320.10 1.68 to the questioning of the viability Total 191 630.03 of this model as an overall useful predictor. And while the ANOVA information does lead to the conclusion that this model is useful 1 to make Treasury Bill Security rate predictions using the Federal Funds Rate, as indicated by the low P-value; the overall ability of the Federal Funds Rate to predict the One Year Treasury Security rate requires further study. H 0: The Model is not useful and contains no linear relation.
H a: The Model is useful and has linear relation.
Multiple Linear Regression: Using the Federal Funds Rate, Standard and Poor s 500, and the Average Selling Price of Housing in the United States to predict the yield to maturity on a One Year Treasury Security. H 0 : ß 1 : The Federal Funds Rate does not contribute to the model consisting of all other predictors. H 0 : ß 2 : The Standard and Poor s 500 does not contribute to the model consisting of all other predictors. H 0 : ß 3 : Average Selling Price of a US Home does not contribute to the model consisting of all other predictors. Figure 2: MiniTab Output for Multiple Linear Regression. The regression equation is Tbill_MnthlyPctChng = - 0.00020 + 0.0963 FFR_Pctchng - 0.00024 SNP_PctChng_Monthly - 0.0005 Housing_AvgPctChng Predictor Coef SE Coef T P Constant -0.000197 0.008724-0.02 0.982 Fed Funds Rate 0.09633 0.05022 1.92 0.057 S & P 500-0.000237 0.001153-0.21 0.838 Avg. Selling Price -0.00046 0.01753-0.03 0.979 S = 0.0623407 R-Sq = 2.0% R-Sq(adj) = 0.4% MiniTab was next used to predict the monthly percentage change in the One Year Treasury Security using the percentage changes in the Federal Funds Rate, the Standard and Poor s 500, and the Average Selling Price (ASP) of a US Home. Output confirmed the previous simple linear regression; that there is indeed some relationship between the Federal Funds Rate and the One Year Treasury Security. In this regression the relatively low P-value for the Federal Funds Rate indicates that it does contribute to the model which includes the S&P 500 and the ASP of a US Home. However the exceptionally high P-values of both the S & P 500 and the ASP of a US Home indicate they do not contribute to the overall model. The coefficients for the slope of the lines of the S & P 500 and the ASP of a US Home are in direct conflict with the slope of the Federal Funds Rate. Analysis of Variance Source DF SS MS F P Regression 3 0.014777 0.004926 1.27 0.287 Residual Error 188 0.730636 0.003886 Total 191 0.745413 Residuals for the Multiple Linear Regression confirm (despite the one outlier in the Versus Fits scatter plot) that this model is not useful; ANOVA further solidifies this confirmation. The large P-value stemming from the ANOVA F-test indicates weak evidence against the model null hypothesis 1 ; implying that there is little to no linear relationship between the One Year Treasury Security and the combined variables: Federal Funds Rate, S & P 500, and ASP of a US Home.
Chi-Square Test: Comparing the proportions of increasing and decreasing months of the Federal Reserve Federal Funds Rate and the Standard and Poor s 500. H 0 : The direction of movement of the Federal Funds Rate has no association with the direction of movement in the Standard and Poor s 500. H a : The direction of movement of the Federal Funds Rate has an association with the direction of movement of the Standard and Poor s 500. Since the previous Multiple Linear Regression analysis indicated that there seemed to be no linear relationship between the One Year Treasury Security and both the S & P 500 and the ASP of a US Home; MiniTab was used to perform a Chi-Square test to determine whether or not there was any association between the movement (increasing or decreasing) of the Federal Funds Rate and the movement of the S & P 500. For the Federal Funds Rate, all the months that had a positive rate change were labeled as FFR Inc, and subsequently all months that had a negative rate change were labeled as FFR Dec. The same process was then applied to the S & P 500 to create categorical variables associated with a numerical change. The Chi-Squared Test led to the conclusion that there is no association between the movement of the Federal Funds Rate and the S & P 500. The P-value returned by the Chi-Square Test indicates very weak evidence against the null hypothesis, leading to its acceptance. Figure 4: MiniTab Output for two One Sample T-tests. Rows: S & P 500 Columns: Fed Funds Rate FFR Dec FFR Inc One Sample and Two Sample T-tests: Testing the movement of the average Federal Funds Rate. Since the Federal Funds Rate can be used to estimate the One Year Treasury Security, it is important to determine whether certain snippets of time have an average rate that is different from the overall Federal Funds Rate. If the mean Federal Funds Rate moves, then there would be periods where the return on One Year Treasury Securities would yield higher or lower returns as compared to the overall average. All SNP Dec 34 33 67 33.50 33.50 67.00 SNP Inc 62 63 125 62.50 62.50 125.00 All 96 96 192 96.00 96.00 192.00 Cell Contents: One-Sample T 1990 s (January 1991 to December 1999) Test of mu = 4.11224 vs not = 4.11224 N Mean StDev SE Mean 95% CI T P 108 4.817 3.674 0.354 (4.116, 5.518) 1.99 0.049 One-Sample T 2000 s (January 2000 to December 2006) Test of mu = 4.11224 vs not = 4.11224 N Mean StDev SE Mean 95% CI T P 84 3.206 4.121 0.450 (2.312, 4.101) -2.01 0.047 Figure 3: MiniTab Output for Chi-Square Test. Count Expected count Pearson Chi-Square = 0.023, DF = 1, P-Value = 0.880 Likelihood Ratio Chi-Square = 0.023, DF = 1, P-Value = 0.880
H 01 : The Federal Funds Rate of the 1990 s is no different than the overall Federal Funds Rate. H a1 : The Federal Funds Rate of the 1990 s is different from the overall Federal Funds Rate. H 02 : The Federal Funds Rate of the 2000 s is no different than the overall Federal Funds Rate. H a2 : The Federal Funds Rate of the 2000 s is different from the overall Federal Funds Rate. As the One Sample T-tests show, in both cases the sample means are different from the overall averages. The P-value of 0.049 as returned by the 1990 s One Sample T-test indicates strong evidence against H 01. In this case the T-test confirms that the Federal Funds Rate in the 1990 s is not the same as the overall Federal Funds Rate from January 1991 to December 2006. In fact, with 95% certainty, the mean can be found somewhere between 4.116% and 5.518% as compared to the overall average of 4.112%. The second One Sample T-test indicates that as well, the mean Federal Funds Rate during January 2000 and December 2006 is not the same as the overall Federal Funds Rate. The P-value of 0.047 is strong evidence against H 02 leading to the conclusion that this mean rate has also moved. Again, with 95% certainty, the mean can be found somewhere between 2.312% and 4.101%. Figure 5: MiniTab Output for a Two Sample T-test. Two-Sample T-Test and CI 1990 s vs 2000 s Sample N Mean StDev SE Mean 1 108 4.82 3.67 0.35 2 84 3.21 4.12 0.45 Difference = mu (1) - mu (2) Estimate for difference: 1.611 95% CI for difference: (0.481, 2.740) T-Test of difference = 0 (vs not =): T-Value = 2.82 DF = 167 P-Value = 0.005 Finally, since there was no overlapping confidence interval between the 1990 s and the 2000 s, a Two Sample T-test was run to determine if there was a difference in the two means. The test confirmed that yes there is a difference in the two means, as indicated by the small P- value. This is strong evidence against the null hypothesis, leading to a rejection of the null, and a conclusion that the means are indeed different. A 95% confidence interval puts the difference somewhere between 0.481% and 2.740%. H 0 : The mean Federal Funds Rate of the 1990 s is not different from the mean Federal Funds Rate in the 2000 s. H a : The mean Federal Funds Rate of the 1990 s is different from the mean Federal Funds Rate in the 2000 s.
Data Sources Bureau of Labor Statistics. Consumer Price Index - All Urban Consumers. US Department of Labor. http://data.bls.gov/servlet/surveyoutputservlet?data_tool=latest_numbers&series_id=cusr0000sa0&ou tput_view=pct_1mth (10 November 2008). Federal Reserve Board of Governors. Monthly Federal Funds Rate. http://www.federalreserve.gov/releases/h15/data/monthly/h15_ff_o.txt (1 December 2008). Federal Reserve Bank. One Year Government Securities by Month. http://federalreserve.gov/releases/h15/data/monthly/h15_tb_y1.txt (4 December 2008). Office of Federal Housing Enterprise Oversight. Housing Prices Indexes. http://www.ofheo.gov/media/hpi/monthlyindex_to_1991.xls (8 November 2008). The National Data Book. Banking, Finance, & Insurance: Money Stock, Interest Rates, Bond Yields. US Census Bureau. http://www.census.gov/compendia/statab/tables/08s1167.xls (10 November 2008). The Federal Reserve Board. H.15 Selected Interest Rates. Federal Reserve Bank. https://www.federalreserve.gov/datadownload/download.aspx?rel=h15&series=40afb80a445c5903ca2c4 888e40f3f1f&filetype=csv&label=include&layout=seriescolumn&from=01/01/1990&to=12/31/2008 (10 November 2008). Yahoo! Finance. S&P 500 Historical Index. http://finance.yahoo.com/q/hp?s=%5egspc&a=00&b=1&c=1990&d=11&e=31&f=2006&g=m (16 November 2008).