Simple Linear Regression | Introduction To Financial Python on QuantConnect

# Simple Linear Regression | Introduction To Financial Python on QuantConnect > Source: https://www.quantconnect.com/learning/articles/introduction-to-financial-python/simple-linear-regression Simple Linear Regression | Introduction To Financial Python on QuantConnect Pricing Data Community Algorithm Lab Documentation Sign In learning center articles / Introduction To Financial Python Simple Linear Regression 9/14 Author Jing Wu 2018-06-09 Introduction In finance and economics filed, most of the models are linear ones. We can see linear regression everywhere, from the foundation of the model portfolio theory to the nowadays popular Fama-French asset pricing model. It's very important to understand how linear regression works in order to have a comprehensive understanding of those theories. If we are holding a stock, we must be curious about the relationship between our stock return and the market return. Let's say we hold Amazon stock on the first day of this year. In order to see the relation directly, we plot the daily return of our stock on the y-axis and plot the S&P 500 index daily return on the x-axis. from datetime import datetime qb = QuantBook() tickers = ["AMZN", "SPY"] symbols = [qb.AddEquity(ticker).Symbol for ticker in tickers] # Get stock prices history = qb.History(symbols, datetime(2017, 1, 1), datetime(2017, 6, 30), Resolution.Daily) df = np.log(history.close.unstack(0)).diff().dropna() df.columns = tickers df.tail() Time AMZN SPY 2017-06-24 0.002434 0.001193 2017-06-27 -0.009771 0.000658 2017-06-28 -0.017456 -0.008089 2017-06-29 0.013777 0.008911 2017-06-30 -0.014647 -0.008828 We successfully create a DataFrame contains the daily logarithm return of Amazon stock and S&P500. Now let's plot it: import matplotlib.pyplot as plt plt.figure(figsize = (15,10)) plt.scatter(df.SPY,df.AMZN) plt.show() The plot is scattered, but we can see they are approximately correlated: generally the higher SPX's daily return is, the higher Amazon stock's return is. This is called positively correlated . We will cover it in the following tutorials. Slope and Intercept It's natural that we want to model the relation between these two rates of return. Intuitively we use a straight line to model it, this is called Linear Regression . In order to find the best straight line, it's natural to think that the vertical distances between the points of the data set and the fitted line should be minimized. Those vertical distances are called residual . Our objective is to make the sum of squared residuals as small as possible. This method is called ordinary least square , or OLS method. We use x and y to represent the two variable, S&P 500 daily returns and AMZN daily returns. The linear relation is: Where is called intercept , is called slope . Generally, if the scatter points can be represented by , then the intercept and slope are given by: Where is the mean of X, is the mean of Y. In python, we don't need to do the above calculation manually because we have package for it. But it still very important to understand the calculation process of in order the understand the modern portfolio theory and CAPM, which we will cover in the future. Python Implementation In python, we have a very power package for mathematical models, which is named 'statsmodels'. import statsmodels.formula.api as sm model = sm.ols(formula = 'AMZN~SPY',data = df).fit() print(model.summary()) We built a simple linear regression model above by using the ols() function in statsmodels. The 'model' instance has lots of properties. The most commonly used one is parameters, or slope and intercept. We can access to them by: print(f'pamameters: {model.params}') [out]: pamameters: Intercept 0.001504 SPY 0.937181 dtype: float64 print(f'residual: {model.resid.tail()}') [out]: residual: time 2017-06-24 -0.000189 2017-06-27 -0.011892 2017-06-28 -0.011379 2017-06-29 0.003921 2017-06-30 -0.007879 dtype: float64 print(f'fitted values: {model.predict()}') [out]: fitted values: [ 7.06344221e-03 7.59665647e-04 4.85148106e-03 -1.59418889e-03 ... ] Now let's have a look at our fitted line: plt.figure(figsize = (15,10)) plt.scatter(df.SPY,df.AMZN) plt.xlabel('spx_return') plt.ylabel('amzn_return') plt.plot(df.SPY,model.predict(),color = 'red') plt.show() The red line is the fitted linear regression straight line. As we can see there are lot of statistical results in the summary table. Now let's talk about some important statistical parameters. Parameter Significance From the summary table we can see 'std err'. This means standard errors of the intercept and slope. The null hypothesis here is , and the alternative hypothesis is . This hypothesis score is calculated by: Where SE is given by: The distribution used here is 'Student's t-distribution'. It's different from normal distribution but used in the similar way. The column 't' in this table is the test score, and 'p>|t|' is the p-value. By observing the p-value, we can see that the significance level of spy, or the slope, is very high because the p-value is close to zero. In other words, we have 99.999 confidence to claim that the slope is not 0, and there exists linear relation between X and Y. However, regarding the intercept, the p-value is 0.923, which means we have only 7.7% confidence level that the value of intercept is not 0. We can also see from the plot that the line crosses the origin. The following 2 columns are the lower band and upper band of the parameters at 95% confidence interval. At 95% confidence level, we can claim that the true value of the parameter is within this range. Model Significance Sum of Squared Errors, or SSE , is used to measure the difference between the fitted value and the actual value. It's given by: If the linear model perfectly fitted the sample, the SSE would be zero. The reason we use squared error here is that the positive and negative errors would offset each other if we simply summed them up. Another measurement of the dispersion of the sample is called total sum of squares, or SS. . it's given by: If you are familiar with variance, we can see that SS divided by the number of sample n is the sample variance. From SSE and SS, we can calculate the Coefficient of Determination , or r-square for short. R-square means the proportion of variation that 'explained' by the linear relationship between X and Y, it's calculated by: Let's assume that the model perfectly fitted the sample, which means all of the sample points lie on the straight line. Then the SSE would become zero, and the r-square would become 1. This means perfect fitness. The higher r-square is, the more parts of variation can be explained by the linear relation, the higher significance level the model is. Some other parameters, such as F-statistic, AIC and BIC, are related to multiple linear regression, with would be cover in the next chapter. Summary In this chapter we introduced how to implement simple linear in python, and focused on how to read the summary table. In next chapter we will introduced multiple linear regression, which are commonly used to built models in finance and economics. 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Simple Linear Regression | Introduction To Financial Python on QuantConnect Source: Simple Linear Regression | Introduction To Financial Python on QuantConnect Pricing Data Community Algorithm Lab Documentation Sign In learning center articles / Introduction To Financial Python Simple Linear Regression 9/14 Author Jing Wu 2018-06-09 Introduction In finance and economics filed, most of the models are linear ones. We can see linear regression everywhere, from the foundation of the model portfolio theory to the nowadays popular Fama-French asset pricing model. It's very important to understand how linear regression works in order to have a comprehensive understanding of those theories. If we are holding a stock, we must be curious about the relationship between our stock return and the market return. Let's say we hold Amazon stock on the first day of this year. In order to see the relation … Ứng dụng: nối nghiên cứu với programming, USD, lãi suất và risk regime — đưa vào journal và playbook. DOI/OA chỉ là rail tham chiếu; nội dung chính là summary, takeaways và ứng dụng thị trường.