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Statistical Methods in Economics

Notes σ
b yx = r y
σ x
When deviations are taken from actual means of X and Y

Σ xy
b yx = Σ x 2

When deviations are taken from assumed means of X and Y

N Σ dd y – ( x x )Σ ( d d y ) Σ
b yx =
N Σ 2 – ( x d x )Σ d 2

Calculating Correlation from Regression Coefficients
It should be interesting to note that the underroot of the product of the two regression coefficients
gives us the value of the coefficient of correlation. Symbolically:

r = b 1 × b or b xy ×b yx
2
σ
Proof: b or b xy = r σ x y
1

σ y
b or b yx = r σ x
2

σ σ y
b 1 × b 2 = r σ x y × r σ x = r 2

∴ r = b 1 × b 2
Since the value of the coefficients of correlation (r) cannot exceed one, one of the regression coefficients
must be less than one, or in other words, both the regression coefficients cannot be greater than one.
×
For example, if b yx = 1.2 and b xy = 1.4, r would be 1.2 1.4 = 1.29 which is not possible. Further,
the regression coefficient which may exceed one should also be such in value that when multiplied
by the other coefficient the underroot of the product of the two coefficients does not exceed one. Also
both the regression coefficients will have the same sign, i.e., they will be either positive or negative.
The coefficient of correlation (r) will have the same sign as that of regression coefficients, i.e., if
regression coefficients have a negative sign, r will also have negative sign and if regression coefficients
have a positive sign, r will also have positive sign.
For example, if b xy = – 0.8 and b yx = – 1.2, r would be

×
–0.8 –1.2 = – 0.98
σ x
Since b xy = r σ y

we can find out any of the four values, given the other three. For example, if we know that r = 0.6, σ x
= 4 and σ xy = 0.8, we can find σ y

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