PSR

Partial-sample regression function to estimate the similarity, informativeness, and relevance of dependent variables.

Description

Estimate results from the partial-sample regression model as described by Czasonis, Kritzman, and Turkington in their 2020 research paper (Journal of Portfolio Management, see reference link below).

Addition by Subtraction: A Better Way to Forecast Factor Returns (and Everything Else)The Journal of Portfolio Management

Journal of Portfolio Management research paper: Addition by Subtraction (Partial Sample Regression)

One of our principals, Mark Kritzman, introduces this powerful model in a lecture at State Street's research retreat in 2020. View a recording on the lecture below.

Forecasting Technique with Applications to Factors, Correlations, and Elections | State Street Corporation

Partial Sample Regression Model in Practice (Recorded Lecture)

Syntax

The following describes the function signature for use in Microsoft Excel's formula bar.

=PSR(whichStat, y, x, theta)
=PSR(whichStat, y, x, theta, )

Input(s)

Argument	Description
whichStat	Required. String to specify statistic to return, use one of the following options: "similarity" "informativeness" "relevance" "scaledrelevance" "rank" "filter" = dummy vector to indicate relevant cross-sectional observations "weighted", "relevanceweighted" "yhat" = forecast value(s) for the dependent variable.
y	Required. Time series or matrix of dependent variables. This is typically the time series of your portfolios, managers, or asset class returns.
x	Required. Time series or matrix of independent variables. This is typically a set of economic variables or factors.
theta	Vector of predictor values, $\hat{x}$ , to use with the model parameters (coefficients) to forecast the response variable $\hat{y}$ . If this argument is empty, the function will assume the most recent cross-sectional values of the independent variables.
threshold	Optional. Relevance threshold, numerical value to specify the minimum percentage of or relevant periods. If the argument is not specified, it defaults to 0.50 (at least 50% relevant periods will be included in the forecast of the partial-sample regression).

Name-Value Optional Arguments

Specify optional pairs of arguments where Name is the option argument name and Value is the corresponding input object. Name-value arguments must appear after other input arguments above, but the order of these pairs does not matter.

Example:

=PSR(whichStat, y, x, theta, "Name1", value1, "Name2", value2, ..., "NameN", valueN) 

Name	Value
threshold	Threshold value to determine relevance cutoff. If the argument is not specified, it defaults to 0.50 (at least 50% relevant periods will be included in the forecast of the partial-sample regression). See also isPercentile option.
isPercentile	Logical, to indicate whether the threshold value is in percentile units or a level value, default = true.
thresholdDirection	Value to indicate the criteria to evaluate relevance against the threshold value set $value \in\begin{cases} -1: & < \\+1: & \geq \end{cases}$ The default threshold direction is $\geq$ .
solveMaxFit	Logical (TRUE or FALSE) flag. If true, the regression model will solve for the maximum fit.
selectVariables	Logical (TRUE or FALSE) flag. If true, the regression model will solve for maximum fit with the optimal selection of variables. If false, then the model will use all variables when solving for maximum fit.
covariance	Covariance matrix of the independent variables.

Output(s)

The function's output will vary depending on the specification of the whichStat argument. The following table will describe the corresponding output result. For M-dependent variables (y) and N-independent variables (x) across T-observations:

whichStat	Output
yHat, prediction	Forecast of dependent variable(s) from the partial sample regression model.
relevance	Tx1 vector of relevance scores. Relevance is the sum of statistical similarity and informativeness. I.e. Relevance is a measure of the importance of an observation to prediction. Its components are the informativeness of past circumstances, the informativeness of current circumstances, and the similarity of past circumstances to current circumstances.
similarity	Tx1 vector of statistical similarity, measured as the negative of the Mahalanobis distance of the past observations for the independent variables to the current values for the independent variables. Or put simply, past observations that are like the current observations are more relevant.
informativeness	Tx1 vector of informativeness as measured by the Mahalanobis distance of the historical observations of the independent variables from its average values.
infoTheta	Tx1 vector of informativeness as measured by the Mahalanobis distance of the historical observations of the independent variables from the circumstances specified (theta).
weights	Tx1 Vector of partial sample regression weights.
fit	1xM Fit values. Fit is the average alignment between relevance and outcomes across all observation pairs for a single prediction. A large value indicates that the observations that are similarly relevant have similar outcomes, in which case on should have more confidence in the prediction. A small value indicates that relevance does not line up with the outcomes, in which case one should view the prediction more cautiously.
filter, included	Tx1 Dummy vector to indicate sub-sample periods that meet the threshold criteria.

Example

Identify similarity, informativeness, and relevance - deepen your regression models

37KB

PSR.xlsx

Example Workbook: PSR