<p>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. </p>

<ul>
<li>Take a look at <a href="http://www.econ.nyu.edu/user/nadiri/pub86.PDF" rel="nofollow noreferrer">Nadir and Prucha</a>'s modelling approach.</li>
<li>In chapter 7 of <a href="https://www.amazon.co.uk/Introductory-Econometrics-Practical-Hamid-Seddighi/dp/0415566878" rel="nofollow noreferrer">this book</a>, the authors develop an accelerator model of investment. You may then calculate the depreciation rate using the parameter estimates of the model.</li>
</ul>

CC BY-SA 3.0
2018-01-16T04:18:20.887Z
check this out: http://www.statcan.gc.ca/pub/15-206-x/15-206-x2015039-eng.htm
EconJohn
CC BY-SA 3.0
2018-01-12T21:38:24.083Z
2018-01-12T21:38:24.083Z
london
<p>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 $K_0$</p>

<p>$$K_1 = (1-\delta)K_0 + I_1$$</p>

<p>$$K_2 = (1-\delta)K_1 + I_2$$</p>

<p>etc.
But how do we do that without specifying $\delta$?</p>

<p>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?"</p>

<p>--</p>

<p>Solving backwards the law of motion of capital we get</p>

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

<p>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 </p>

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

<p>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 $\delta$ </p>

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

<p>through some iterative algorithm.</p>

CC BY-SA 3.0
2018-01-14T00:12:16.913Z
Hmmm.. Sounds like an akward way to get an estimate. Can you elaborate on how you would do that?
EconJohn
CC BY-SA 3.0
2018-01-14T22:04:01.043Z
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.
london
CC BY-SA 3.0
2018-01-12T22:19:09.497Z
2018-01-14T22:28:06.877Z
2018-01-14T22:28:06.877Z
Alecos Papadopoulos
<p>In an econometric sense, how do macroeconomists measure capital depreciation $\delta$ in the formula $K_{t+1}=K_t(1-\delta)+I_{t+1}$.</p>

<p>What data would I need and what kind of regression would I run?</p>

<macroeconomics><applied-econometrics>
Macroeconometrics:How to measure capital depreciation?
EconJohn
# Macroeconometrics:How to measure capital depreciation?

(Asked by: username_0 on 2018-01-12)

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?

**Tags:** macroeconomics, applied-econometrics


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## 2 answers

### Answer by username_1 on 2018-01-12. Score: 2

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](http://www.econ.nyu.edu/user/nadiri/pub86.PDF)'s modelling approach.
* In chapter 7 of [this book](https://www.amazon.co.uk/Introductory-Econometrics-Practical-Hamid-Seddighi/dp/0415566878), the authors develop an accelerator model of investment. You may then calculate the depreciation rate using the parameter estimates of the model.

> **Comments:**
> **username_0**: check this out: http://www.statcan.gc.ca/pub/15-206-x/15-206-x2015039-eng.htm  

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### Answer by username_2 on 2018-01-12. Score: 1

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?"

--

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.

> **Comments:**
> **username_0**: Hmmm.. Sounds like an akward way to get an estimate. Can you elaborate on how you would do that?  
> **username_1**: 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.  

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