CFAI Level 2, Book 1, SS11, Q.17, p.463

CFA Level II Quant
#1

Can someone please explain the answer to this question? I don’t understand how the natural log subtractions in this equation simplify.

The question is: Suppose we decide to use and autoregressive model with a seasonal lag because of the seasonal autorcorrelation. We are modeling quarterly data, so we estimate the equation to be: (ln Salest - ln Salest-1) = b0 + b1(ln Salest-1 - ln Salest-2) + b2(ln Salest-4 - ln Salest-5) +et. Where: b0 = 0.0121; b1 = -0.0839; b2 = 0.6292 If sales grew by 1 percent last quarter and by 2 percent four quarters ago, use the model to predict the sales growth for this quarter.

Where I get stuck is the answer, How does the above equation (with coefficients filled in):

(ln Salest - ln Salest-1) = 0.0121 - 0.0839(ln Salest-1 - ln Salest-2) + 0.6292(ln Salest-4 - ln Salest-5)

Transform in to this equation?:

(ln Salest - ln Salest-1) = 0.0121- 0.0839ln(1.01) + 0.6292ln(1.02) = e0.02372 - 1 = 2.40%

I’m not sure I follow the “ln” logic in this equation, could someone walk me through the throught process on how everything cancels out?

#2

You need to use logarithmic algebra:

Ln[sales(x) - sales(y)]=Ln[sales(x)/sales(y)]

If sales(x)=1.01[sales(y)], then sales(x)/sales(y)=1.01

That should sort things out for you.

#3

Thank you ayousaf!

Now, just so I can check my understanding… when we get the equation down to the following form:

(ln Salest - ln Salest-1) = 0.02372 this can then be expressed as:

ln[(Salest)/(Salest-1)] = 0.02372, and then we take the antilog to get to:

[(Salest)/(Salest-1)] = e^(0.02372) --> [(Salest)/(Salest-1)] = 1.02400, or that

Salest = 1.024Salest-1, which is essentially saying that no matter what the value of Salest-1, we will see an increase of 2.4% of that number. The problem for me is that Salest-1 is just apparently dropped, I think I’m still missing something.

Also, they don’t work the solution out this way, instead they show the answer as: e^(0.02372) -1. Which to me means that they are cancelling variables to get the 1, but I can’t see how.

I think this is the same question asked two ways, but for some reason this problem is really tripping me up, so any help you could give would help a lot.

#4

No they’re not cancelling any variables out. According to your OP:

y = (ln Salest - ln Salest-1) = 0.0121- 0.0839ln(1.01) + 0.6292ln(1.02)

y = 0.0121 - 0.0839(0.01) + 0.6292(0.02)

y = 0.0121 - 0.0008 + 0.0125

y = 0.0238 (difference due to rounding)

So, to get this in non-logarithmic terms:

e^y = e^0.0238 = 1.0241

Then you subtract 1 from this to get the percentage = 0.0241 = 2.41%

#5

In case this helps to see where the “-1” term comes from…

e^(ln(a/b)) = a/b where a is the final value and b is the initial value.

So, in this example where y-hat is the predicted Ln[Sales(t)/Sales(t-1)]:

e^(y-hat) = [Sales (t) / Sales (t-1)]

To see the percent increase, use final minus initial, all divided by initial:

[Sales(t) - Sales (t-1)] / [Sales (t-1)] = [Sales (t) / Sales (t-1)] -1

Again, we’re subtracting 1 to see the percent increase.

#6

Thank you both, I see the issue clearly now.