1998issue C031-5
Evaluating linear regression baselines for index valuation
A linear specification can treat an equity index as the joint outcome of several estimated influences, then be checked for fit, coefficient signs, and significance across sample windows. Editorial interpretation: use that sequence as an evaluation instrument, and reserve attention for large residual extremes rather than minor gaps or timing claims.
- A linear specification can treat an equity index level as the joint outcome of several estimated influences, so any single correlation is not a finished specification.
- R-squared measures how much index variation is associated with the chosen inputs, but a high reading remains suspect if theoretically relevant influences are omitted.
- Significance testing starts from the presumption of no relationship and rejects that presumption only at a preselected error probability.
- Fitted levels describe valuation given contemporaneous inputs and help flag large residual extremes. They do not forecast when prices will change.
A joint baseline, not a single factor
A linear specification can treat an equity index level as the dependent outcome of several jointly estimated influences rather than of a single factor.
Single-factor and combined regressions of candidate influences against the S&P 500 each showed some correlation, so specification work had to go beyond detecting any association.
Fit statistics are not a complete check
Fit statistics such as R-squared measure how much index variation is associated with the chosen inputs, while a high reading can still be suspect if theoretically relevant influences are omitted.
The retained specification used trailing-year earnings, the dividend payout ratio, and the five-year Treasury yield as the joint inputs to an estimated S&P 500 level.
Across three sample windows and frequencies, the joint equations posted R-squared values of 96%, 96%, and 93%.
Coefficient signs can reject a busy equation
Entering earnings and dividends together produced very low or negative earnings coefficients because the two series are related. Replacing dividends with the payout ratio removed that peculiarity.
Editorial interpretation: an equation that detects association is not finished if the estimated signs contradict the intended roles of the inputs. Coefficient signs belong in the evaluation, not after it.
Significance starts from no relationship
Significance testing starts from the presumption that the inputs do not relate to the index and requires a preselected error probability before that presumption is rejected.
Equations were evaluated at a 0.05% significance level, corresponding to a 99.95% probability threshold for rejecting no relationship.
The first three decade windows from 1948 through 1977 failed the significance test, whereas windows from 1977 through 1996 and the full 49-year span met it.
Editorial interpretation: meeting the threshold in one window is not the same as meeting it in every window. Demand the check outside the most convenient sample before treating the baseline as settled.
Fitted levels describe now, not when
The fitted levels describe valuation given contemporaneous inputs and do not forecast when prices will change. They are presented as adjuncts for spotting large residual extremes rather than minor gaps.
All readings on this track · 43 readings
- 1990Constructing dollar baselines from rates, inflation, and residuals
- 1990Constructing a nominal index value from forward earnings and fitted yield
- 1990Constructing a nominal index price from earnings and a fitted yield
- 1990Endpoint-pinned price paths are not forecasts
- 1990Constructing least-squares polynomial smoothers
- 1991Endpoint growth rates versus linear-regression consistency
- 1991Out-of-sample checks for linear growth fits
- 1991Trend as persistence, not a straight line
- 1991Quadratic trend, residual oscillator, and a secondary cycle calendar
- 1991Time-origin offset and residual-price divergence on a quadratic least-squares fit
- 1991A least-squares trendline from ordered prices
- 1992Constructing log-linear growth and reliability screens
- 1992Constructing log-linear growth-rate baselines
- 1992Next-session high, low, and close from rolling linear regression
- 1993Auditing an index price-earnings multiple with short-rate regression
- 1994Regression-seeded nested exponential price filter
- 1994Constructing the double exponential average from lag cancellation
- 1994Evaluating money supply as a linear leading-index baseline
- 1995Constructing least-squares trend channels
- 1995Linear baseline holdout checks for annual bill-rate forecasts
- 1995Projection bands from high and low regression slopes
- 1995Evaluating a least-squares end-point moving average on a known test series
- 1996Constructing an endpoint moving average from a least-squares line
- 1996Scoring equity path consistency with a k-ratio overlay
- 1996Evaluating month-end yield gaps for equity regimes
- 1996Constructing session-indexed standard error bands
- 1998Evaluating linear regression baselines for index valuation
- 1998R-squared as a two-state trend filter from a price-time fit
- 2000Second-order moving-average lag correction
- 2002Price regression line versus beta for index tracking
- 2003Regression slope with an r-squared trend confidence gate
- 2003Constructing finite-volume-element divergence with slope comparison
- 2004Building a daily score from regression, retracement, and volume
- 2004Constructing least-squares trendlines from ordered prices
- 2007Rectangle breakout targets beyond height
- 2007Confirming a price trend with regression slope and r-squared
- 2008A linear-regression angle assembled as one trend filter
- 2010A two-state swing machine from four running extremes
- 2016Score oil-complex tightness before divergence or regression
- 2017Nikkei-yen intermarket divergence as a regime case study
- 2017Constructing Calmar ratio and linear regression baselines
- 2019Pair-trade layer construction versus average-spread management
- 2020A convolution slope built from nested linear regression