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):

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.