PCA Factors Describe Returns but Do Not Forecast Them on Their Own

# Forecasting next day return of a stock using PCA of index constituents


# Forecasting next day return of a stock using PCA of index constituents












I am trying to predict the return of BN4.SI ( a singapore stock ) and part of Strait Times 30 component index using principal component Analysis. I have written my code in python.

My Question is i have got factors loading how can i predict the next day return of BN4.SI based on this PCA Factors.

Please help me.

The steps i have taken are

- Get the matrix of standardised returns of all stocks. Standardised log returns are [(x - mean)/std for x in array]

- Generate covariance matrix of the return matrix and that should give me a square matrix ( m X m)

- I am using numpy linear algebra eig function to calculate eigen values and eigen vectors. Since my returns are standardised then sum of eigen values should be equalled to number of components.

- Then sort the eigen values in decending order to get first 3 eigen values which explains almost 80% of the variance. This is shown in scree plot in the picture added below.

- Next step is to get top 3 PCA vector which is acheived by getting the dot product of eigen vector and the original return matrix data. pca1 = ti.np.dot(newDF,eVector1.reshape(-1,1)).reshape(1,-1) pca2 = ti.np.dot(newDF,eVector2.reshape(-1,1)).reshape(1,-1) pca3 = ti.np.dot(newDF,eVector3.reshape(-1,1)).reshape(1,-1)

- To perform linear regression with BN4.SI standarised return data. I need to get the matrix of the transpose of PCA vectors. np.column_stack([pca1.T,pca2.T,pca3.T])

- I am using sklearn to do linear regression. from sklearn import linear_model model = linear_model.LinearRegression() model.fit(ti.np.column_stack([pca1.T,pca2.T,pca3.T]),newDF["BN4.SI"]) model.score(ti.np.column_stack([pca1.T,pca2.T,pca3.T]),newDF["BN4.SI"]) print(model.coef_) print(model.intercept_) [ 0.2802088 0.37944899 0.13462393] -1.15582251414e-18

## Answer by Richi Wa (score 0)

https://quant.stackexchange.com/a/34665

I decided to write an answer in order not to write too many comments.

What do we get by PCA? Let us assume we have $n$ random variables. We get a representation of the data of the form $$ X_i = e_{1,i} P_1 + \cdots + e_{n,i} P_n $$ where the PCAs $P_1,\ldots,P_n$ are uncorrelated (not necessarily independent) and ordered (descending) by variance.

The above relation is estimated from past data.

In the world of interest rates $X_i$ could be the change (rather not the level) of vertex $i$ and it turns out that the change of the i-th rate can be decomposed into a parallel shift ($P_1$) and a steppening ($P_2$) and so on. This conditional on the future realization of $P_1$ we can estimate the change of vertex $i$ as it is estimated to be proportional by a factor $e_{1,i}$ to the parallel shift. The relation above does not tell us whether $P_{1,t+1}$ the future pf $P_1$ will be positive or negative.

In the case of stocks we usually take $X_i$ to be the return of asset $i$. And the principle components are a market factor and several long/short portfolios. If we can predict the market then we can plug this into the above equation.

Note that as far as I remember $e_{1,i}$ is not the beta of the stock in the stock setting. Thus we would get a forecast cheaper by looking a betas and forecasting the broad market.