import yfinance as yf
import numpy as np
import pandas as pd
import matplotlib
import matplotlib.pyplot as pltWorkshop 3, Econometric Models
Abstract
This is an INDIVIDUAL workshop. In this workshop we learn more about linear regression and the Capital Asset Pricing Model. The CAPM can be estimated by running a regression model with premium returns.
1 Estimating the CAPM model for a stock
2 The CAPM model
The Capital Asset Pricing Model states that the expected return of a stock is given by the risk-free rate plus its beta coefficient multiplied by the market premium return. In mathematical terms:
E[R_i] = R_f + β_1(R_M − R_f )
We can express the same equation as:
(E[R_i] − R_f ) = β_1(R_M − R_f )
Then, we are saying that the expected value of the premium return of a stock is equal to the premium market return multiplied by its market beta coefficient. You can estimate the beta coefficient of the CAPM using a regression model and using continuously compounded returns instead of simple returns. However, you must include the intercept b0 in the regression equation:
(r_i − r_f ) = β_0 + β_1(r_M − r_f ) + ε
Where ε ∼ N(0, σ_ε); the error is a random shock with an expected mean=0 and a specific standard deviation or volatility. This error represents the result of all factors that influence stock returns, and cannot be explained by the model (by the market).
In the market model, the dependent variable was the stock return and the independent variable was the market return. Unlike the market model, here the dependent variable is the difference between the stock return minus the risk-free rate (the stock premium return), and the independent variable is the premium return, which is equal to the market return minus the risk-free rate. Let’s run this model in with a couple of stocks.
3 Data collection
We load the libraries to collect, process and visualize stock data from Yahoo Finance:
3.1 Download stock data
We download monthly stock data for Apple, Tesla and the S&P500 from Dec 2019 to July 31, 2024 from Yahoo Finance using the yfinance function and obtain continuously compounded returns for each:
data = yf.download("^GSPC AAPL TSLA", start='2020-01-01', end='2025-01-31', interval='1mo', auto_adjust=True)
[ 0% ]
[**********************67%******* ] 2 of 3 completed
[*********************100%***********************] 3 of 3 completed# Get adjusted close prices
adjprices = data['Close']
# Calculate continuously compounded returns for the 3 prices:
#returns = np.log(adjprices) - np.log(adjprices.shift(1))
#returns = returns.dropna()
returns = np.log(adjprices).diff(1).dropna()
# I used the diff function instead of subtracting the log price and its previous log priceI have monthly returns from Jan 2020:
returnsTicker AAPL TSLA ^GSPC
Date
2020-02-01 -0.124201 0.026424 -0.087860
2020-03-01 -0.069944 -0.242781 -0.133668
2020-04-01 0.144424 0.400210 0.119421
2020-05-01 0.078963 0.065730 0.044287
2020-06-01 0.140190 0.257109 0.018221
2020-07-01 0.152834 0.281421 0.053637
2020-08-01 0.194233 0.554719 0.067719
2020-09-01 -0.106370 -0.149762 -0.040018
2020-10-01 -0.061888 -0.100372 -0.028056
2020-11-01 0.089481 0.380309 0.102146
2020-12-01 0.110196 0.217731 0.036449
2021-01-01 -0.005517 0.117344 -0.011199
2021-02-01 -0.084562 -0.161038 0.025757
2021-03-01 0.008806 -0.011270 0.041563
2021-04-01 0.073453 0.060293 0.051097
2021-05-01 -0.053514 -0.126372 0.005471
2021-06-01 0.096197 0.083548 0.021971
2021-07-01 0.062958 0.010974 0.022493
2021-08-01 0.040114 0.068224 0.028578
2021-09-01 -0.068965 0.052633 -0.048738
2021-10-01 0.057001 0.362230 0.066858
2021-11-01 0.098461 0.027238 -0.008369
2021-12-01 0.073061 -0.079968 0.042689
2022-01-01 -0.015837 -0.120597 -0.054018
2022-02-01 -0.056856 -0.073397 -0.031863
2022-03-01 0.057156 0.213504 0.035148
2022-04-01 -0.102177 -0.213125 -0.092068
2022-05-01 -0.057505 -0.138340 0.000053
2022-06-01 -0.083469 -0.118657 -0.087652
2022-07-01 0.172804 0.280480 0.087201
2022-08-01 -0.033093 -0.075250 -0.043367
2022-09-01 -0.127556 -0.038314 -0.098049
2022-10-01 0.103956 -0.153347 0.076835
2022-11-01 -0.035243 -0.155866 0.052358
2022-12-01 -0.128762 -0.457813 -0.060782
2023-01-01 0.104829 0.340916 0.059921
2023-02-01 0.021393 0.171905 -0.026459
2023-03-01 0.113647 0.008471 0.034451
2023-04-01 0.028575 -0.233184 0.014536
2023-05-01 0.043647 0.216022 0.002479
2023-06-01 0.091525 0.249689 0.062719
2023-07-01 0.012704 0.021392 0.030664
2023-08-01 -0.044658 -0.035588 -0.017875
2023-09-01 -0.091510 -0.030929 -0.049946
2023-10-01 -0.002573 -0.219832 -0.022225
2023-11-01 0.106443 0.178464 0.085424
2023-12-01 0.014808 0.034390 0.043279
2024-01-01 -0.043145 -0.282704 0.015771
2024-02-01 -0.019992 0.075015 0.050428
2024-03-01 -0.051373 -0.138383 0.030547
2024-04-01 -0.006729 0.041725 -0.042506
2024-05-01 0.121059 -0.028782 0.046904
2024-06-01 0.092614 0.105428 0.034082
2024-07-01 0.052982 0.159378 0.011258
2024-08-01 0.030684 -0.080549 0.022578
2024-09-01 0.018473 0.200441 0.019996
2024-10-01 -0.030902 -0.046071 -0.009946
2024-11-01 0.049315 0.323147 0.055720
2024-12-01 0.054788 0.157011 -0.025308
2025-01-01 -0.059308 0.001880 0.0266583.2 Download risk-free data from the FED
We download the risk-free monthly rate for the US (3-month treasury bills), which is the TB3MS ticker. We do this with the pandas_datareader library:
# You have to install the pandas-datareader package:
#!pip install pandas-datareader
import pandas_datareader.data as pdrimport pandas_datareader.data as pdr
import datetime
# I define start as the month Jan 2020
start = datetime.datetime(2020,1,1)
# I define the end month as July 2024
end = datetime.datetime(2023,12,31)
Tbills = pdr.DataReader('TB3MS','fred',start,end)We see the content of Tbills:
Tbills TB3MS
DATE
2020-01-01 1.52
2020-02-01 1.52
2020-03-01 0.29
2020-04-01 0.14
2020-05-01 0.13
2020-06-01 0.16
2020-07-01 0.13
2020-08-01 0.10
2020-09-01 0.11
2020-10-01 0.10
2020-11-01 0.09
2020-12-01 0.09
2021-01-01 0.08
2021-02-01 0.04
2021-03-01 0.03
2021-04-01 0.02
2021-05-01 0.02
2021-06-01 0.04
2021-07-01 0.05
2021-08-01 0.05
2021-09-01 0.04
2021-10-01 0.05
2021-11-01 0.05
2021-12-01 0.06
2022-01-01 0.15
2022-02-01 0.33
2022-03-01 0.44
2022-04-01 0.76
2022-05-01 0.98
2022-06-01 1.49
2022-07-01 2.23
2022-08-01 2.63
2022-09-01 3.13
2022-10-01 3.72
2022-11-01 4.15
2022-12-01 4.25
2023-01-01 4.54
2023-02-01 4.65
2023-03-01 4.69
2023-04-01 4.92
2023-05-01 5.14
2023-06-01 5.16
2023-07-01 5.25
2023-08-01 5.30
2023-09-01 5.32
2023-10-01 5.34
2023-11-01 5.27
2023-12-01 5.24The TB3MS serie is given in percentage and in annual rate. I divide it by 100 and 12 to get a monthly simple rate since I am using monthly rates for the stocks:
rfrate = Tbills / 100 / 12Now I get the continuously compounded return from the simple return:
rfrate = np.log(1+rfrate)I used the formula to get cc reteurns from simple returns, which is applying the natural log of the growth factor (1+rfrate)
4 Visualize the relationship
We do a scatter plot putting the S&P500 premium returns as the independent variable (X) and Tesla premium return as the dependent variable (Y). We also add a line that better represents the relationship between the stock returns and the market returns:
import seaborn as sb
plt.clf()
x = returns['GSPC_Premr']
y = returns['TSLA_Premr']
# I plot the (x,y) values along with the regression line that fits the data:
sb.regplot(x=x,y=y)
plt.xlabel('Market Premium returns')
plt.ylabel('TSLA Premium returns')
plt.show()
Sometimes graphs can be deceiving. In this case, the range of X axis and Y axis are different, so it is better to do a graph where we can make both X and Y ranges with equal distance.
plt.clf()
sb.regplot(x=x,y=y)
# I adjust the scale of the X axis so that the magnitude of each unit of X is equal to that of the Y axis
plt.xticks(np.arange(-0.8,1,0.2))([<matplotlib.axis.XTick object at 0x00000171F9940190>, <matplotlib.axis.XTick object at 0x00000171F98F39D0>, <matplotlib.axis.XTick object at 0x00000171F9942FD0>, <matplotlib.axis.XTick object at 0x00000171F9943750>, <matplotlib.axis.XTick object at 0x00000171F9943ED0>, <matplotlib.axis.XTick object at 0x00000171F997C690>, <matplotlib.axis.XTick object at 0x00000171F997CE10>, <matplotlib.axis.XTick object at 0x00000171F997D590>, <matplotlib.axis.XTick object at 0x00000171F997DD10>], [Text(-0.8, 0, '−0.8'), Text(-0.6000000000000001, 0, '−0.6'), Text(-0.40000000000000013, 0, '−0.4'), Text(-0.20000000000000018, 0, '−0.2'), Text(-2.220446049250313e-16, 0, '0.0'), Text(0.19999999999999973, 0, '0.2'), Text(0.3999999999999997, 0, '0.4'), Text(0.5999999999999996, 0, '0.6'), Text(0.7999999999999996, 0, '0.8')])# I label the axis:
plt.xlabel('Market Premium returns')
plt.ylabel('TSLA Premium returns')
plt.show()
QUESTION: WHAT DOES THE PLOT TELL YOU? BRIEFLY EXPLAIN
WHAT DOES THE PLOT TELL YOU? BRIEFLY EXPLAIN
R: I CAN SEE THAT THE RANGE OF PREMIUM RETURNS OF THE MARKET (GSPC) RANGES APPROX. BETWEEN -18% AND +8%, WHILE THE TESLA PREMIUM RETURNS RANGES BETWEEN MORE THAN -40% TO +40% MONTHLY RETURNS.
I CAN ALSO SEE THAT TESLA PREMIUM RETURNS ARE POSITIVELY RELATED TO THE MARKET PREMIUM RETURNS, AND ITS SENSITIVITY TO CHANGES IN THE MARKET PREMIUM RETURN IS VERY HIGH. I CAN SEE THAT APPROX. FOR EACH + 1.00 PERCENT POINT OF INCREASE IN THE MARKET PREMIUM RETURN, TESLA PREMIUM RETURN MOVES IN MUCH MORE THAN +1.00 PERCENT POINT SINCE THE SLOPE OF THE LINE IS VERY INCLINED WITH A DEGREE HIGHER THAN 45 DEGREES (INCLINATION OF 45 DEGREES HAPPENS WHEN THE SLOPE=1).
5 Estimating the CAPM model for a stock
Use the premium returns to run the CAPM regression model for each stock.
We run the CAPM for TESLA:
import statsmodels.formula.api as smf
# I estimate the OLS regression model:
mkmodel = smf.ols('TSLA_Premr ~ GSPC_Premr',data=returns).fit()
# I display the summary of the regression:
print(mkmodel.summary()) OLS Regression Results
==============================================================================
Dep. Variable: TSLA_Premr R-squared: 0.392
Model: OLS Adj. R-squared: 0.378
Method: Least Squares F-statistic: 28.98
Date: jue., 06 mar. 2025 Prob (F-statistic): 2.55e-06
Time: 15:50:52 Log-Likelihood: 19.958
No. Observations: 47 AIC: -35.92
Df Residuals: 45 BIC: -32.22
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 0.0205 0.024 0.862 0.393 -0.027 0.068
GSPC_Premr 2.2322 0.415 5.383 0.000 1.397 3.067
==============================================================================
Omnibus: 0.470 Durbin-Watson: 1.636
Prob(Omnibus): 0.791 Jarque-Bera (JB): 0.495
Skew: -0.220 Prob(JB): 0.781
Kurtosis: 2.757 Cond. No. 17.6
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.I can important results of the model in the following variables:
b0=mkmodel.params.iloc[0]
b1=mkmodel.params.iloc[1]
seb0 = mkmodel.bse.iloc[0]
seb1 = mkmodel.bse.iloc[1]
tb0 = mkmodel.params.iloc[0] / mkmodel.bse.iloc[0]
tb1 = mkmodel.params.iloc[1] / mkmodel.bse.iloc[1]
pvalueb0 = mkmodel.pvalues.iloc[0]
pvalueb1 = mkmodel.pvalues.iloc[1]
confb0 = 100 * (1-pvalueb0)
minb0 = mkmodel.conf_int().iloc[0].iloc[0]
maxb0 = mkmodel.conf_int().iloc[0].iloc[1]
minb1 = mkmodel.conf_int().iloc[1].iloc[0]
maxb1 = mkmodel.conf_int().iloc[1].iloc[1]The beta0 coefficient of the model is 0.0205, while beta1 is 2.2322.
The 95% confidence interval for beta0 goes from -0.0274 to 0.0683, while the 95% confidence interval for beta1 goes from 1.397 to 3.0674.
If we subtract and add about 2 times the standard error of beta0 to beta0 we get the 95% confidence interval for beta0. Why? Because thanks to the Central Limit Theorem, beta0the beta coefficients will behave similar to a normal distributed variables since the beta0 can be expressed as a linear combination of random variables.
We can construct the 95% confidence interval for beta1 in the same way we calculate the 95% C.I. for beta0.
6 CHALLENGE 1
Respond the following questions regarding Tesla CAPM model:
(a) INTERPRET THE RESULTS OF THE COEFFICIENTS (b0 and b1), THEIR STANDARD ERRORS, P-VALUES AND 95% CONFIDENCE INTERVALS.
REGARDING BETA0:
BETA0 IS 0.0205 . REMEMBER THAT BETA0 IN THE CAPM REPRESENTS THE EXCESS PREMIUM RETURNS OF THE STOCK COMPARED TO THE PREMIUM RETURNS OF THE MARKET.
THEN, THE VALUE ESTIMATED FOR BETA0 IS ACTUALLY THE AVERAGE VALUE OF BETA0. THEN, ON AVERAGE IT IS POSITIVE.
HOWEVER, IS BETA0 SIGNIFICANTLY POSITIVE? IN OTHER WORDS, IN THE FUTURE, IS IT PROBABLE THAT BETA0 WILL KEEP BEING POSITIVE 95% OF THE TIME?
WE CAN RESPOND THIS QUESTION ABOUT THE STATISTICAL SIGNIFICANCE OF BETA0 WITH ANY OF THE FOLLOWING:
| PARAMETER | RULE OF THUMB FOR SIGNIFICANCE (95% CONFIDENCE) |
|---|---|
| t-VALUE | IF |t-VALUE| > 2 |
| P-VALUE | IF |P-VALUE|<0.05 |
| 95% CONFIDENCE INTERVAL | IF THE INTERVAL DOES NOT CONTAINS THE NUMBER ZERO |
ACCORDING TO THE 95% CONFIDENCE INTERVAL, IN THE FUTURE BETA0 CAN MOVE FROM -0.0274 TO 0.0683 WITH A PROBABILITY OF 0.95. THIS MEANS THAT, ALTHOUGH BETA0 ON AVERAGE IS POSITIVE, IT IS NOT SIGNIFICANTLY POSITIVE AT THE 95% CONFIDENCE INTERVAL. IN OTHER WORDS, WITH A PROBABILITY OF 0.95, IN THE FUTURE BETA0 CAN BE NEGATIVE, ZERO OR POSITIVE! WE HAVE NOT ENOUGH CERTAINTY AT 95% CONFIDENCE TO SAY THAT BETA0 IS POSITIVE.
SINCE BETA0 CAN MOVE FROM A NEGATIVE NUMBER TO A POSITIVE NUMBER, ITS PVALUE IS GREATER THAN 0.05 (0.393). THE PVALUE IS THE PROBABILITY OF MAKING A MISTAKE IF WE REJECT THE NULL HYPOTHESIS, WHICH STATES THAT BETA0=0. IN OTHER WORDS, (1-PVALUE) WILL BE THE PROBABILITY THAT WE WILL BE CORRECT IF WE REJECT THE NULL HYPOTHESIS AND ACCEPT THE ALTERNATIVE HYPOTHESIS THAT STATES THAT BETA IS GREATER THAN ZERO. IN THIS CASE, WE HAVE A PROBABILITY OF 0.607 IF WE SAY THAT THE EXCESS RETURN OF TESLA WILL BE POSITIVE IN THE FUTURE.
THE t VALUE IS LESS THAN 2. REMEMBER THAT THE t-VALUE IS THE DISTANCE FROM THE BETA0 COEFFICIENT TO THE ZERO MEASURED IN STANDARD DEVIATIONS OF BETA0. THEN, IF t-VALUE < 2, THIS MEANS THAT THE DISTANCE OF BETA0 FROM THE ZERO IS NOT ENOUGH TO SAY THAT IT WILL BE POSITIVE 95% OF THE TIME.
REMEMBER THAT IN THE CAPM REGRESSION MODEL BETA0 REPRESENTS EXCESS PREMIUM RETURNS OF THE STOCK OVER THE MARKET PREMIUM RETURNS. IN THIS CASE, ON AVERAGE TESLA HAS EXCESS PREMIUM RETURNS OVER THE PREMIUM RETURNS OF THE MARKET. HOWEVER, WE CANNOT SAY THAT TESLA OFFERS EXCESS (POSITIVE) PREMIUM RETURNS OVER THE MARKET AT THE 95% CONFIDENCE. THIS IS IN LINE WITH THE MARKET EFFICIENCY HYPOTHESIS, WHICH PREDICTS THAT BETA0 WILL NOT BE SIGNIFICANTLY DIFFERENT THAN ZERO FOR ANY STOCK WITHIN A FINANCIAL MARKET.
REGARDING BETA1:
BETA1 IS 2.2322 . REMEMBER THAT BETA1 IN THE CAPM REPRESENTS THE SENSITIVITY OF THE STOCK PREMIUM RETURNS WITH RESPECT TO THE MARKET PREMIUM RETURNS.
IN THE CAPM BETA1 ACTUALLY MEASURES TWO IMPORTANT ASPECTS OF THE RELATIONSHIP BETWEEN THE STOCK AND THE MARKET:
THE LINEAR RELATIONSHIP BETWEEN THE STOCK PREMIUM RETURNS AND THE MARKET PREMIUM RETURNS
THE MARKET RISK OF THE STOCK
BET1 IS ACTUALLY THE THE SLOPE (INCLINATION) OF THE REGRESSION LINE.
THEN, THE VALUE ESTIMATED FOR BETA1 FOR TESLA IS ACTUALLY THE AVERAGE VALUE OF BETA1. THEN, ON AVERAGE IT IS POSITIVE AND IT IS GREATER THAN 1.
HOWEVER, IS BETA1 SIGNIFICANTLY POSITIVE? IN OTHER WORDS, IN THE FUTURE, IS IT PROBABLE THAT BETA1 WILL KEEP BEING POSITIVE 95% OF THE TIME?
THE OTHER INTERESTING QUESTION FOR BETA1 IS:
IS BETA1 SIGNIFICANTLY GREATER THAN 1? IN OTHER WORDS, IN THE FUTURE, IS IT PROBABLE THAT BETA1 WILL KEEP GREATER THAN 1 95% OF THE TIME? SINCE BETA1 REPRESENTS MARKET RISK OF THE STOCK, WE CAN ASK: IS TESLA SIGNIFICANTLY RISKIER THAN THE MARKET?
WE CAN RESPOND THE FIRST QUESTION WHETHER BETA1 IS SIGNIFICANTLY POSITIVE WITH ANY OF THE FOLLOWING:
| PARAMETER | RULE OF THUMB FOR SIGNIFICANCE (95% CONFIDENCE) |
|---|---|
| t-VALUE | IF |t-VALUE| > 2 |
| P-VALUE | IF |P-VALUE|<0.05 |
| 95% CONFIDENCE INTERVAL | IF THE INTERVAL DOES NOT CONTAINS THE NUMBER ZERO |
ACCORDING TO THE 95% CONFIDENCE INTERVAL, IN THE FUTURE BETA1 CAN MOVE FROM 1.397 TO 3.0674 WITH A PROBABILITY OF 0.95. THIS MEANS THAT, IT IS SIGNIFICANTLY POSITIVE AT THE 95% CONFIDENCE INTERVAL. IN OTHER WORDS, WITH A PROBABILITY OF 0.95, IN THE FUTURE, TESLA PREMIUM RETURNS WILL BE POSITIVELY RELATED TO THE MARKET PREMIUM RETURNS.
SINCE BETA1 CAN MOVE FROM A POSITIVE NUMBER TO ANOTHER POSITIVE NUMBER, ITS PVALUE IS LESS THAN 0.05 (0). THE PVALUE IS THE PROBABILITY OF MAKING A MISTAKE IF WE REJECT THE NULL HYPOTHESIS, WHICH STATES THAT BETA1=0. IN OTHER WORDS, (1-PVALUE) WILL BE THE PROBABILITY THAT WE WILL BE CORRECT IF WE REJECT THE NULL HYPOTHESIS AND ACCEPT THE ALTERNATIVE HYPOTHESIS THAT STATES THAT BETA1 IS GREATER THAN ZERO. IN THIS CASE, WE HAVE A PROBABILITY OF 0.607 IF WE SAY THAT THE BETA1 OF TESLA WILL BE POSITIVE IN THE FUTURE.
THE t VALUE IS GREATER THAN 2. REMEMBER THAT THE t-VALUE IS THE DISTANCE FROM THE BETA1 COEFFICIENT TO THE ZERO MEASURED IN STANDARD DEVIATIONS OF BETA1. THEN, IF t-VALUE < 2, THIS MEANS THAT THE DISTANCE OF BETA1 FROM THE ZERO IS ENOUGH TO SAY THAT IT WILL BE POSITIVE 95% OF THE TIME.
THE SECOND QUESTION ASKS WHETHER BETA1 IS SIGNIFICANTLY GREATER THAN ONE. WE CAN RESPOND THIS QUESTION WITH THE 95% CONFIDENCE INTERVAL, BUT NOT WITH THE p-VALUE NOR THE t-VALUE CALCULATED IN THE REGRESSION! IN THIS CASE, SINCE THE 95% CONFIDENCE INTERVAL DOES NOT CONTAIN THE NUMBER 1, THEN WE CAN SAY THAT TESLA WILL BE SIGNIFICANTLY RISKIER THAN THE MARKET SINCE ITS BETA1 WILL BE GREATER THAN 1 95% OF THE TIME!
(b) ACCORDING TO THE EFFICIENT MARKET HYPOTHESIS, WHAT IS THE EXPECTED VALUE OF b0 in the CAPM REGRESSION MODEL?
ACCORDING TO THE EFFICIENT MARKET HYPOTHESIS, THE EXPECTED VALUE OF BETAO IS ZERO SINCE THE MARKET EFFICIENT HYPOTHESIS STATES THAT THE MARKET IS THE MOST EFFICIENT PORTFOLIO, SO THE MARKET SYSTEMATICALLY OUTPERFORMS ANY INDIVIDUAL STOCK. THEN, FOR ANY STOCK BETA0 SHOULD NOT BE SIGNIFICANTLY POSITIVE.
IN REAL FINANCIAL MARKETS THERE ARE VERY FEW STOCKS THAT IN SOME PERIODS SYSTEMATICALLY OUTPERFORMS THE MARKET. HOWEVER, IT IS COMMON TO SEE THAT THIS IS JUST A TIME WINDOW THAT SOONER OR LATER DISAPEARS.
(c) ACCORDING TO YOUR RESULTS, IS TESLA SIGNIFICANTLY RISKIER THAN THE MARKET ? WHAT IS THE t-test YOU NEED TO DO TO RESPOND THIS QUESTION? Do the test and provide your interpretation. (Hint: Here you have to change the null hypothesis for b1: H0: b1=1; Ha=b1<>1)
AS I MENTIONED EARLIER, THE 95% CONFIDENCE INTERVAL FOR TESLA STARTS IN A NUMBER GREATER THAN +1, SO TESLA IS SIGNIFICANTLY RISKIER THAN THE MARKET.
7 CHALLENGE 2
Follow the same procedure to get Apple’s CAPM and respond the following questions:
(a) INTERPRET THE RESULTS OF THE COEFFICIENTS (b0 and b1), THEIR STANDARD ERRORS, P-VALUES AND 95% CONFIDENCE INTERVALS.
(b) ACCORDING TO THE EFFICIENT MARKET HYPOTHESIS, WHAT IS THE EXPECTED VALUE OF b0 in the CAPM REGRESSION MODEL?
(c) ACCORDING TO YOUR RESULTS, IS TESLA SIGNIFICANTLY RISKIER THAN THE MARKET ? WHAT IS THE t-test YOU NEED TO DO TO RESPOND THIS QUESTION? Do the test and provide your interpretation. (Hint: Here you have to change the null hypothesis for b1: H0: b1=1; Ha=b1<>1)
WHAT IS THE EFFICIENT MARKET HYPOTHESIS? BRIEFLY DESCRIBE WHAT THIS HYPOTHESIS SAYS.
YOU HAVE TO DO YOUR OWN RESEARCH
8 READING
Read carefully: Basics of Linear Regression Models.
9 Quiz 3 and W3 submission
Go to Canvas and respond Quiz 3 about Linear Regression. You will be able to try this quiz up to 3 times. Questions in this Quiz are related to concepts of the readings related to this Workshop. The grade of this Workshop will be the following:
Complete (100%): If you submit an ORIGINAL and COMPLETE HTML file with all the activities, with your notes, and with your OWN RESPONSES to questions
Incomplete (75%): If you submit an ORIGINAL HTML file with ALL the activities but you did NOT RESPOND to the questions and/or you did not do all activities and respond to some of the questions.
Very Incomplete (10%-70%): If you complete from 10% to 75% of the workshop or you completed more but parts of your work is a copy-paste from other workshops.
Not submitted (0%)