Quantitative Methods Flashcards
Covariance
SUM[(Xbar-Xi)*(Ybar-Yi)] / (n-1)
Correlation Coefficient
r = Cov(x,y) / [Sigma(x)*Sigma(y)]
where, -1 < r < 1
T-test
t = [r * sqrt(n-2)] / sqrt(1-r^2)
Reject H0 if t+critical < t or t-critical>t.
t needs to be in between the two critical t’s
Slope of regression line
b1 = Covariance / Variance
Total Sum of Squares (SST)
SUM(Yi - Ybar)^2
Sum of Squared Errors (SSE)
SUM(Yi-Yhat)^2
This is the unexplained variation in the regression
Regression Sum of Squares (RSS)
SUM(Yhat-Ybar)^2
This is the explained variation in the regression
Total Variation = Explained Variation + Unexplained Variation
SST = RSS + SSE
Mean Regression Square of Sums (MSR)
=RSS/number of slope parameters
Mean Squared Error (MSE)
MSE = SSE / (n-2)
Regression degree of freedom = k
Error degree of freedom = n - k - 1
Where k = number of slope parameters
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R^2 = (SST - SSE) / SST = RSS / SST
R^2 = (Total Variation - Unexplained Variation) / Total Variation = Explained Variation / Total Variation
Standard Error of the Estimate (SEE)
SEE = SqRt (MSE) = SqRt(SSE/(n-2))
Smaller SEE means the regression has better fit
F-Test tests the statistical significance of a regression
F = MSR / MSE = (Rss/k)/(SSE/[n-k-1])
Always a one tail test
Reject if F > Fcritical
F-test hypothesis testing
- H0 –> b1=b2=b3=b4=0 vs. Ha–> @ least 1 bi is not =0
- Decision —> Reject H0 if F(test-statistic) > Fcritical
- If rejected, Bj is significant
Adjusted R^2
Ra^2 = 1 - {[(n-1)/(n-k-1)]*(1-R^2)}
t = coefficient / standard error
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Breusch Pagan Chi-Square Test is used to detect heterskedasticity
ChiSquare = n * R^2residuals
with k degrees of freedom.
To correct heteroskedasticity, calculate the robust standard errors.
Durbin-Watson Statistic is used to detect serial correlation (residual terms are correlated with each other)
DW = SUM[(EtHAT - Et-1HAT)^2] / SUM(EtHAT^2)
Decision Rule:
Dw < dl reject null, positive serial correlation
dl du fail to reject null, no evidence of serial correlation
Use the Hansen method to correct for serial correlation
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Detect Multicollinearity 1. t-test says no variable is statistically significant, and 2. F-test is statistically significant and R^2 is high.
Correct multicollinearity by omitting highly correlated variables
Log-Linear Models
Yt = e^(b0+b1(t)) ---> ln(Yt) = b0 + b1(t)
Autoregressive Models
Xt = b0 + b1Xt-1 + b2Xt-2+…+bpXt-p + ErrorTerm
Chain rule of forecasting
Xt+1 = b0 + b1*Xt Xt+2 = b0 + b2*Xt+1
T-test
t = correlation of error terms / [1/SqRt(T)]
with T-2 degrees of freedom.
If the test is rejected, add more lag variables and re-test
Mean Reverting Level is the tendency to move back to the mean
Xt = b0/(1-b1)
Dickey Fuller Test
- Xt = b0 + b1*Xt-1 + Error
- Xt - Xt-1 = b0 + (b1-1)*Xt-1 + Error
- Test the (b1-1) term using the t-test. If it is rejected, there is no unit root.
A unit root is when a time series is not covariance stationary.
Auto Regressive Conditional Heteroskedasticity (ARCH) exists if the variance of the residuals in one period is dependent on the variance of the residuals in a previous period.
Errort^2 = A0 + A1*ErrorT-1^2 + Sigmat
Variance of ARCH series
Sigmat+1^2 = A0 + A*Errort^2
