The visual information on a scatterplot can be greatly enhanced, with little additional cost, by computing and plotting smoothed points. Robust locally weighted regression is a method for smoothing a scatterplot, (x i , y i ), i = 1, …, n, in which the fitted value at z k is the value of a polynomial fit to the data using weighted least squares, where the weight for (x i , y i ) is large if x i is close to x k and small if it is not. A robust fitting procedure is used that guards against deviant points distorting the smoothed points. Visual, computational, and statistical issues of robust locally weighted regression are discussed. Several examples, including data on lead intoxication, are used to illustrate the methodology.

Robust Locally Weighted Regression and Smoothing Scatterplots

Locally weighted regression, or loess, is a way of estimating a regression surface through a multivariate smoothing procedure, fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series With local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions, such as polynomials The goal of this article is to show, through applications, how loess can be used for three purposes: data exploration, diagnostic checking of parametric models, and providing a nonparametric regression surface Along the way, the following methodology is introduced: (a) a multivariate smoothing procedure that is an extension of univariate locally weighted regression; (b) statistical procedures that are analogous to those used in the least-squares fitting of parametric functions; (c) several graphical methods that are useful tools for understanding loess estimates and checking the a

https://www.stat.washington.edu/courses/stat527/s14/readings/Cleveland_Delvin_JASA_1988.pdf

Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting

A linearization of a rational expectations present value model for corporate stock prices produces a simple relation between the log dividend-price ratio and mathematical expectations of future log real dividend changes and future real discount rates. This relation can be tested using vector autoregressive methods. Three versions of the linearized model, differing in the measure of discount rates, are tested for United States time series 1981-1986: versions using real interest rate data. The results yield a metric to judge the relative importance of real dividend growth, measured real discount rates and unexplained factors in determining the dividend-price ratio.

The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors

This paper analyzes the relation of stock volatility with real and nominal macroeconomic volatility, economic activity, financial leverage, and stock trading activity using monthly data from 1857 to 1987. An important fact, previously noted by Officer (1973), is that stock return variability was unusually high during the 1929-1939 Great Depression. While aggregate leverage is significantly correlated with volatility, it explains a relatively small part of the movements in stock volatility. The amplitude of the fluctuations in aggregate stock volatility is difficult to explain using simple models of stock valuation, especially during the Great Depression. ESTIMATES OF THE STANDARD deviation of monthly stock returns vary from two to twenty percent per month during the 1857-1987 period. Tests for whether differences this large could be attributable to estimation error strongly reject the hypothesis of constant variance. Large changes in the ex ante volatility of market returns have important negative effects on risk-averse investors. Moreover, changes in the level of market volatility can have important effects on capital investment, consumption, and other business cycle variables. This raises the question of why stock volatility changes so much over time. Many researchers have studied movements in aggregate stock market volatility. Officer (1973) relates these changes to the volatility of macroeconomic variables. Black (1976) and Christie (1982) argue that financial leverage partly explains this phenomenon. Recently, there have been many attempts to relate changes in stock market volatility to changes in expected returns to stocks, including Merton (1980), Pindyck (1984), Poterba and Summers (1986), French, Schwert, and Stambaugh (1987), Bollerslev, Engle, and Wooldridge (1988), and Abel (1988). Mascaro and Meltzer (1983) and Lauterbach (1989) find that macroeconomic volatility is related to interest rates. Shiller (1981a,b) argues that the level of stock market volatility is too high relative to the ex post variability of dividends. In present value models such as Shiller's, a change in the volatility of either future cash flows or discount rates

/pdf/why-does-stock-market-volatility-change-over-time-1tlvx2plat.pdf

Why Does Stock Market Volatility Change Over Time

A dividend-ratio model is introduced here that makes the log of the dividend-price ratio on a stock linear in optimallyforecastfuture one-period real discount rates andfuture one-period growth rates of real dividends. If expost discount rates are observable, this model can be tested by using vector autoregressive methods. Four versions of the linearized model, differing in the measure of discount rates, are tested for U.S. time series 1871-1986 and 1926-1986: a version that imposes constant real discount rates, and versions that measure discount rates from real interest rate data, aggregate real consumption data, and return variance data. The results yield a metric to judge the relatve importance of real dividend growth, measured real discount rates, and unexplained factors in determining the dividend-price ratio.

/pdf/the-dividend-price-ratio-and-expectations-of-future-2469h44yiz.pdf

A classic collection of raw data and analysis of the movements of interest rates, bond yields and stock prices in the United States between 1856 and 1936. It comments on issues such as leads and lags and empirical forecasting.

Some theoretical problems suggested by the movements of interest rates, bond yields, and stock prices in the United States since 1856

Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856.

The Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856.

The Smoothing of Time Series

A classic collection of raw data and analysis of the movements of interest rates, bond yields and stock prices in the United States between 1856 and 1936 It comments on issues such as leads and lags and empirical forecasting

Frederick R. Macaulay

Papers

Some theoretical problems suggested by the movements of interest rates, bond yields, and stock prices in the United States since 1856

Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856.

The Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856.

The Smoothing of Time Series

Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856