# Macroeconometrics:How to measure capital depreciation?

In an econometric sense, how do macroeconomists measure capital depreciation $\delta$ in the formula $K_{t+1}=K_t(1-\delta)+I_{t+1}$.

What data would I need and what kind of regression would I run?

## 2 Answers

I tried to empirically estimate the rate of depreciation a couple of years ago. I gave up! Following a large literature on growth, I assumed a constant rate of depreciation eventually. I did not have data for all the variables needed to estimate the model.

• Take a look at Nadir and Prucha's modelling approach.
• In chapter 7 of this book, the authors develop an accelerator model of investment. You may then calculate the depreciation rate using the parameter estimates of the model.

In order to measure the depreciation rate from the law of motion of capital, you need to have data on capital. But usually capital data series are constructed from investment, with a more-or-less arbitrary initial capital level (which, by the way, necessitates to discard a portion of the initial values). So we do, for some $K_0$

$$K_1 = (1-\delta)K_0 + I_1$$

$$K_2 = (1-\delta)K_1 + I_2$$

etc. But how do we do that without specifying $\delta$?

One way is to calibrate it rather than estimate it, by using the estimated "capital/output" ratio, which is obtained independently and not using the capital series. Then essentially you ask : "what is the value of $\delta$ so that the implied capital series through the law of motion of capital gives me capital/output ratios that agree with estimated such ratios?"

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Solving backwards the law of motion of capital we get

$$K_t = I_{t-1} + (1-\delta)I_{t-2}+(1-\delta)^2 I_{t-2}+...+ (1-\delta)^tI_0 + (1-\delta)^tK_0$$

For large enough series (40 or more observations, given that the depreciation range, being an average, is not expected to exceed $0.1$, and it is usually computed as less), one ignores the last term involving the initial capital stock which is usually unknown. Assume more over that we have an independent estimate that

$$K_t/Y_t = \bar k_y(t) \implies K_t = \bar k_y(t)Y_t$$

or that we have some "average" accepted value for the capital/output ratio. So, given the gross investment series, output and this average capital/output ratio value, we have to solve for $\delta$

$$\bar k_y(t)Y_t = I_{t-1} + (1-\delta)I_{t-2}+(1-\delta)^2 I_{t-2}+...+ (1-\delta)^tI_0$$

through some iterative algorithm.

• Hmmm.. Sounds like an akward way to get an estimate. Can you elaborate on how you would do that?
– EconJohn
Jan 14 '18 at 0:12
• I think you should be able to do this using Excel's solver function or any programming language. Or, simply solve the following equation for $\delta$: capital/output ratio at time $t$ = $\frac{(1-\delta)K_{t-1}+I_t}{GDP_t}$. You will need data for investment, capital stock, GDP and capital/output ratio. Jan 14 '18 at 22:04