Question:

Estimating Stock Beta Using the Kalman Filter

Nova: 26 June 2022

This is a basic question about using the KalmanFilter function. Lets say I have the following:

enter image description here

I want to use the KalmanFilter function (or other related Kalman Filter functions in Mathematica) to estimate the time-varying beta(t) in the linear model:

AAPL (t) = beta (t) * SP500 (t) + e (t)

Where e(t) is a random error process.

I have tried the following:

AAPL = FinancialData["AAPL", "Return", {2020, 1, 2}];
SP500 = FinancialData["GSPC", "Return", {2020, 1, 2}];
tds = TemporalData[{AAPL, SP500}];
tsm = TimeSeriesModelFit[tds] // Quiet;
Kf = KalmanFilter[tsm];
DateListPlot[Kf]

which produces the following:

enter image description here

But it's not the result I am looking for.

I am unclear as to how to formulate the problem in a way that will produce estimates of beta(t) using Mathematica's KalmanFilter function (or other Kalman-related functions).

Any assistance appreciated.

Thanks!

Answer:
Naomi: 26 June 2022

This isn't a complete answer to your question, in as much as it doesn't use the WL KalmanFilter function. Perhaps someone on the forum can suggest how to adapt this code to make use of the KF functions in Mathematica....

KalmanTimeLoop[Y_, X_, a_, b1_, Ve_, Vw_, Covb1_] := 
 Module[{Yhat, b, k, n, e, t, Q, K, Covb, Zscore},
  (************************************************************) 
  (************************************************************) 
  (** UPDATES THE KALMAN FILTER PARAMETERS FOR EACH TIMESTEP ***)
  (************************************************************) \

  (*                                                        
  X = k x n matrix of k prices data series for t = 1,...,n ;
      note that X(t)[[1,All]] = 1 for all t;
  a = k x k State Transition matrix estimated in initialization;
  b = k x n matrix of state vectors b(t) = {b1,...,bk},
  Ve = scalar price error variance, estimated in initialization;
  Vw = k x k state noise covariance matrix, 
  estimated in initialization;
  Covb[[t]] = k x k state covariance matrix at time t = 1,...,n;
  *)
  (***********************************************************) 

  {k, n} = Dimensions@X;
  e = ConstantArray[0, n];
  Q = ConstantArray[0, n];
  b = ConstantArray[0, {k, n}];
  Covb = ConstantArray[0, {n, k, k}];
  b[[All, 1]] = b1;
  Covb[[1]] = Covb1;
  Do[
   (*************************************************)
   (* 
   update state prediction and state covariance matrix*)

   If[t > 1, b[[All, t]] = a . b[[All, t - 1]];
    Covb[[t]] = a . Covb[[t - 1]] . Transpose[a] + Vw;
    Covb[[t]] = DiagonalMatrix@Diagonal@Covb[[t]];];
   (*************************************************)
   (* 
   Measurement Prediction Equation *)

   Yhat = X[[All, t]] . b[[All, t]];
   (*************************************************)
   (* 
   Measurement Prediction Error *)
   e[[t]] = Y[[t]] - Yhat;
   (*************************************************)
   (* 
   Measurement Covariance Prediction *)     
   Q[[t]] = X[[All, t]] . Covb[[t]] . X[[All, t]] + Ve; 
   (*************************************************)
    (* 
   Kalman Gain *)                  
   K = Covb[[t]] . X[[All, t]]/Q[[t]];  
   (*************************************************) 
   (* State Update *)
   b[[All, t]] = b[[All, t]] + K*e[[t]];
   (*************************************************) 
   (* State Covariance Update *)                  
   Covb[[t]] = 
    Covb[[t]] - DiagonalMatrix[K*X[[All, t]]] . Covb[[t]]; 
   (*************************************************) 
   , {t, 1, n}];
  Zscore = e/Sqrt[Q];
  {b, Covb, K, e, Q, Zscore}]