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A Framework for Projecting Interest Rate Spreads and Volatilities January 2000

As noted earlier, the Congressional Budget Office's (CBO's) projections of alternative interest rates used a statistical model in which spreads were estimated as a function of four factors: rates on three-month Treasury bills, the spread between a long-term and a short-term interest rate, the level of inflation, and the volatility of the interest rate on federal funds.⁽¹⁾ Each projection equation had the same set of explanatory variables, but each yielded a different set of coefficients.
 

THE MODEL

CBO's methods for estimating the model drew from standard econometric theory in selecting the appropriate form of the spread and the techniques to be applied. The following equations and the succeeding discussion incorporate a summary of those methods.

The spread between the alternative interest rates and the rate on three-month Treasury bills was modeled as follows:

Equation 1.     

where for R_j,t, the upper case R signifies an interest rate (expressed as a bond-equivalent yield) and the subscript j stands for one-month commercial paper, three-month commercial paper, one-month London interbank offer rate, three-month LIBOR, one-month Eurodollar, and three-month Eurodollar, respectively; "ln" signifies the natural logarithm of the expression within braces; the two variables preceded with an upper case D signify (for reasons described later) that the variable takes on a value of 1 in the year and quarter (1974Q3 and 1980Q4) and zero otherwise; and the variable u_j,t represents unobservable random factors, or residuals.

In equation 1, the spread appears in ratio form to the left of the equal sign. It is expressed as the logarithm of the ratio of 1 plus the interest rate under examination to 1 plus the rate on three-month Treasury bills. That form of the spread is approximately equal to the numerical difference between interest rates when rates are low. CBO analysts chose it to help reduce the effect of extreme values of the spread on the statistical estimates of parameters (a_j,0, ....., a_j,6). (For example, it reduces peak values of the spread between one-month LIBOR and three-month Treasury bill rates by about 40 to 50 basis points compared with that spread constructed as the simple difference between the two rates.) Even so, the two extreme values of the spread (which occurred in the third quarter of 1974 and the fourth quarter of 1980) were still too large to be accounted for by the determining factors despite the logarithmic transformation. As a result, two "dummy" variables, D74Q3 and D80Q4, were introduced to help remove the effects of those two extreme values on the parameter estimates. Omitting the dummy variables would alter the projections of the spreads. Most important, it would significantly increase the uncertainties associated with them and therefore the estimated probability of the rates' exceeding any given threshold.

Equation 2 is the part of the structural model dealing with nonuniform variations in the volatility of the residuals of the spread--for example, variation during the period before the mid-1980s, when inflation was climbing and rates were high, compared with variation since then, when inflation has declined and rates are low:

Equation 2.      ln{u²_j,t} = b_j,0 + b_j,1 ln{u²_j,t-1} + b_j,2 ln{1 + Inflation_t}

Nonuniform volatility of the residuals introduces heteroskedasticity, which is characterized in part by a periodic clustering of large values. That type of time variation giving rise to nonuniformity is known as autoregressive-conditional heteroskedasticity, or ARCH.

Accounting for ARCH, as in equation 2, improves empirical estimates of the parameters of equation 1 over longer data intervals, such as the nearly 30-year period encompassing the 1970s through the 1990s. Not accounting for ARCH could mean that the potential range of variation around the estimated values of the parameters in equation 1 would be too large to yield much confidence in the estimates themselves. An unattractive alternative would be to shorten the period of analysis--unattractive because a shorter period, such as the 1990s alone or the late 1980s to late 1990s, limits understanding of how the determining factors affect the spread.

Including ARCH also helps analysts develop reasonable probability ranges over the projection span. In particular, as specified in equation 2, the low inflation of the 1990s appears to have played an important role in reducing volatility. As such, a baseline projection of low inflation in the years ahead leads to projections of spreads with less volatility than might otherwise be the case. Less volatility in the spreads in turn produces probability ranges that are narrower than they might otherwise have been.
 

THE ESTIMATING PROCEDURE

CBO's estimating procedure is a sequence of three steps, which make up the method of generalized least squares. In the first step, equation 1 uses ordinary least squares to derive an estimate of the unobservable series of residuals u_j,t (see Table A-1). The difference between the actual value of the spread and the value predicted by the equation yields the residuals used in the second step.
 

In that step, the residuals obtained from the first equation are squared to obtain a measure of the spread's volatility. That measure is then used along with the inflation rate to estimate equation 2 for each of the six spreads (see Table A-2). The projected volatility values are retrieved from this equation for the last step of generalized least squares.
 

TABLE A-2. CBO ESTIMATES OF SQUARED RESIDUALS USING THE ARCH MODEL
Determinants		Commercial Paper			LIBOR			Eurodollar Deposits
		1-Month	3-Month		1-Month	3-Month		1-Month	3-Month
Constant		-12.018	-12.506		-10.266	-9.644		-8.718	-9.009
(t-Stat)		(-8.454)	(-9.071)		(-7.371)	(-7.764)		(-6.727)	(-7.454)
ln(u2-1)		.167	.149		.207	.295		.370	.343
(t-Stat)		(1.762)	(1.596)		(2.168)	(3.315)		(4.174)	(3.948)
ln(1 + Inflation)		21.485	22.825		8.545	22.208		18.758	21.218
(t-Stat)		(2.799)	(2.691)		(0.859)	(3.222)		(2.150)	(3.136)
Measures of Fit
	R-bar squared	.108	.076		.040	.216		.200	.242
	Standard error (In basis points)	2.153	2.469		2.856	1.909		2.463	1.886
	F (2,106)	7.54	5.47		3.225	15.87		14.48	18.26
SOURCE: Congressional Budget Office.
NOTES: The equation was used to estimate the model. The estimation interval extends from the third quarter of 1971 (1971Q3) to the third quarter of 1998 (1998Q3).
ARCH = autoregressive-conditional heteroskedasticity; LIBOR = London interbank offer rate.

In the third step, the square root of the projected volatility value is divided into all of the observable variables of equation 1, and the resulting equation is estimated by ordinary least squares. Of course, having divided each observation by the predicted volatility from equation 2 amounts to a reweighting of each observation (see Table A-3). The explicit reweighting of observations is a critical feature of the third step. Without it, each observation would have been weighted equally. With it, observations are weighted inversely to the degree of volatility found in equation 2. (Figures A-1 and A-2 show the weights for each spread.)
 

TABLE A-3. CBO ESTIMATES OF INTEREST RATE SPREADS USING GENERALIZED LEAST SQUARES
Determinantsa		Commercial Paper			LIBOR			Eurodollar Deposits
		1-Month	3-Month		1-Month	3-Month		1-Month	3-Month
Constant		.0032	.0050		.0012	-.0001		-.0001	-.0006
(t-Stat)		(3.919)	(5.857)		(0.754)	(-0.107)		(-0.739)	(-0.521)
ln(1 + Tbill-1)		.0526	-.0016		.0750	.0696		.0866	.0552
(t-Stat)		(4.574)	(-0.147)		(3.625)	(3.653)		(5.691)	(2.998)
ln(1 + Tnote-1)/(1 + Tbill-1)		-.0882	-.0459		-.1851	-.1101		-.1510	-.1222
(t-Stat)		(-4.005)	(-2.177)		(-4.570)	(-2.780)		(-5.737)	(-3.418)
ln(1 + Inflation-1)		.0058	.0055		.0637	.0644		.0394	.0704
(t-Stat)		(0.502)	(0.526)		(3.678)	(3.565)		(2.894)	(4.281)
ln(1 + Fedfunds volatility-1)		-.0166	-.0077		.0199	.0390		.0246	.0404
(t-Stat)		(-3.507)	(-1.910)		(2.371)	(4.639)		(3.146)	(4.915)
D1974Q3		.0135	.0062		.0200	.0244		.0228	.0258
(t-Stat)		(2.263)	(1.254)		(2.700)	(2.175)		(1.972)	(2.311)
D1980Q4		.0047	-.0084		.0106	.0053		.0117	.0065
(t-Stat)		(1.072)	(-1.972)		(1.712)	(0.683)		(1.414)	(0.865)
Measures of Fit
	R-bar squared	.491	.294		.799	.658		.808	.603
	Standard error (In basis points)	31	31		52	52		51	52
Measures of Residual Randomness
	D.W.	1.09	0.99		1.08	1.00		0.96	0.96
	Q(27)	49.4	90.1		130.8	116.3		114.5	100.9
First-Order Autocorrelation Coefficient		.442	.495		.426	.468		.463	.490
SOURCE: Congressional Budget Office.
NOTES: The dependent variable is [ln{(1 + Rate)/(1 + Tbill)}]/, where is the square root of the fitted values from the ARCH model. The estimation interval is from the third quarter of 1971 (1971Q3) to the third quarter of 1998 (1998Q3).
ARCH = autoregressive-conditional heteroskedasticity; LIBOR = London interbank offer rate; D.W. = Durbin-Watson statistic.
a. The determinants are all divided by >t, the square root of the fitted values from the ARCH model.

FIGURE A-1. DATA WEIGHTS USED IN CBO'S MODEL FOR PROJECTING INTEREST RATE SPREADS FOR ONE-MONTH AND THREE-MONTH COMMERCIAL PAPER

SOURCE: Congressional Budget Office.

NOTES: Spreads (the differences between rates) are computed against the rate for three-month Treasury bills. Interest rates have been converted to bond-equivalent yields.

Commercial paper is defined here as short-term debt issued by financial companies.

FIGURE A-2. DATA WEIGHTS USED IN CBO'S MODEL FOR PROJECTING INTEREST RATE SPREADS FOR ONE-MONTH AND THREE-MONTH LONDON INTERBANK DOLLAR DEPOSITS

SOURCE: Congressional Budget Office.

NOTE: Spreads (the differences between rates) are computed against the rate for three-month Treasury bills. Interest rates have been converted to bond-equivalent yields.

a. The rate on London interbank dollar deposits is known as LIBOR--the London interbank offer rate.

THE DATA SOURCES

Listed below are the data used in CBO's model and their sources:

• Commercial Paper Rates. The interest rates paid by financial issuers. CBO used the quarterly average of daily data published by the Federal Reserve and converted to the bond-equivalent yield, or BEY.⁽²⁾ Data for the period before the fourth quarter of 1997 were based on historical statistics gathered by the Federal Reserve Bank of New York. Data for the fourth quarter of 1997 and later were gathered by the Depository Trust Company and reported to the Federal Reserve Bank of New York.
  
  
• London Interbank Offer Rates. Quarterly average of daily data published by the British Bankers' Association and converted to the BEY.
  
  
• Eurodollar Deposit Rates. Quarterly average of daily data published by the Federal Reserve and converted to the BEY.
  
  
• Treasury Bill Rates. Quarterly average of daily data on three-month Treasury bill rates sold in the secondary markets on a discount basis. The average, published by the Federal Reserve, was converted to the BEY.
  
  
• Ten-Year Treasury Note Rates. Quarterly average of daily data on 10-year Treasury notes with constant maturity, as published by the Federal Reserve.
  
  
• Inflation. Annualized quarterly percentage change of the consumer price index for all urban consumers. (The CPI-U is the quarterly average of monthly levels, published by the Bureau of Labor Statistics.)
  
  
• Federal Funds Volatility. The square root of an eight-quarter moving average of the squared deviation of the federal funds rate. Deviation is measured as the difference between log first-difference change of the federal funds rate and the trend of log first-difference change in the rate. Trend is measured as an eight-quarter moving average. CBO used quarterly data based on the average of daily data published by the Federal Reserve.

1. Under the current terms of the student-loan program, payments to lenders are adjusted quarterly on the basis of a quarterly average of weekly yields. CBO thus estimated the model using quarterly data.

2. CBO drew the rates it used for commercial paper, Eurodollar deposits, Treasury securities, and federal funds from Board of Governors of the Federal Reserve, Selected Interest Rates, Federal Reserve Statistical Release H.15 (various dates).

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