which is given (assuming small changes year-on-year) by dividing both sides of this equation by the output, Y: The first two terms on the right hand side of this equation are the proportional changes in labour and capital year-on-year, and the left hand side is the proportional output change. The remaining term on the right, giving the effect of productivity improvements on GDP is defined as the Solow residual: The residual, SR(t) is that part of growth not explicable by measurable changes in the amount of capital, K, and the number of workers, L. If output, capital, and labour all double every twenty years the residual will be zero, but in general it is higher than this: output goes up faster than growth in the input factors. The residual varies between periods and countries, but is almost always positive in peace-time capitalist countries. Some estimates of the post-war U.S. residual credited the country with a 3% productivity increase per-annum until the early 1970s when productivity growth appeared to stagnate. Regression analysis and the Solow residual The above relation gives a very simplified picture of the economy in a single year; what growth theory econometrics does is to look at a sequence of years to find a statistically significant pattern in the changes of the variables, and perhaps identify the existence and value of the "Solow residual". The most basic technique for doing this is to assume constant rates of change in all the variables (obscured by noise), and regress on the data to find the best estimate of these rates in the historical data available (using an Ordinary least squares regression). Economists always do this by first taking the natural log of their equation (to separate out the variables on the right-hand-side of the equation); logging both sides of this production function produces a simple linear regression with an error term, : A constant growth factor implies exponential growth in the above variables, so differentiating gives a linear relationship between the growth factors which can be deduced in a simple regression. In a regression analysis, the equation one would estimate is: where: y is (log) output, ln(Y) k is capital, ln(K) ℓ is labour, ln(L) C can be interpreted as the co-efficient on log(A) – the rate of technological change – (1 − α). Given the form of the regression equation, we can interpret the coefficients as elasticities. For calculation