刘迪:Stata空间溢出效应的动态图形-空间计量

发布时间:2020-10-09 阅读 30

Stata 连享会   主页 || 视频 || 推文

温馨提示: 定期 清理浏览器缓存,可以获得最佳浏览体验。

课程详情 https://gitee.com/arlionn/Course   |   lianxh.cn

课程主页 https://gitee.com/arlionn/Course

This post shows how to create animated graphics that illustrate the spatial spillover effects generated by a spatial autoregressive (SAR) model. After reading this post, you could create an animated graph like the following.

This post is organized as follows. First, I estimate the parameters of a SAR model. Second, I show why a SAR model can produce spatial spillover effects. Finally, I show how to create an animated graph that illustrates the spatial spillover effects.

A SAR model

I want to analyze the homicide rate in Texas counties as a function of unemployment. I suspect that the homicide rate in one county affects the homicide rate in neighboring counties.

I want to answer two questions.

  1. How can I set up a model that explicitly allows the homicide rate in one county to depend on the homicide rate in neighboring counties?
  2. Given my model, if the unemployment rate in Dallas increases to 10%, how would the homicide rate change in the neighboring counties of Dallas ?

Fit a SAR model

A standard linear model for the homicide rate in county i (hratei) as a function of the unemployment rate in that county’s unemploymenti is

A SAR model allows (hratei) to depend on the homicide rate in neighboring counties. I need some new notation to write down a SAR model. I let (Wi,j) be a positive number if county j is a neighbor of county i, zero if the j is not a neighbor of i, and zero if j=i, because no county can border itself.

Given this notation, a SAR model that allows the homicide rate in county i to depend on the homicide rate in neighboring counties can be written as

where (Wi,j) defines the closeness between county i and county j. The term j=1NWi,jhratej is a weighted sum of the homicide rates in county i’s neighboring counties, and it specifies how the homicide rates in neighboring counties affect the homicide rate in county i.

Stacking the neighborhood information in (Wi,j) for each county i produces a matrix W that records the neighbor information for each county i. The matrix W is known as a spatial-weighting matrix.

The spatial-weighting matrix that we are using has a special structure; each element is either a value c or zero, where c is greater than zero. This type of spatial-weighting matrix is known as a normalized contiguity matrix.

In Stata, we use spmatrix to create a spatial-weighting matrix, and we use spregress to fit a cross-sectional SAR model.

I begin by downloading some data on the homicide rates of U.S. counties from the Stata website and creating a subsample that uses only data on counties in Texas.

. /* Get data for Texas counties' homicide rate */
. copy http://www.stata-press.com/data/r15/homicide1990.dta ., replace

. use homicide1990
(S.Messner et al.(2000), U.S southern county homicide rates in 1990)

. keep if sname == "Texas"
(1,158 observations deleted)

. save texas, replace
file texas.dta saved

Intuitively, a file that specifies the borders of all the places of interest is known as a shape file texas.dta is linked to the Stata version of a shape file that specifies the borders of all the counties in Texas. I now download that dataset from the Stata website and use spset to show that they are linked.

. /* Get data for Texas counties' homicide rate */
. copy http://www.stata-press.com/data/r15/homicide1990_shp.dta, replace

. spset
  Sp dataset texas.dta
                data:  cross sectional
     spatial-unit id:  _ID
         coordinates:  _CX, _CY (planar)
    linked shapefile:  homicide1990_shp.dta

I now use spmatrix to create a normalized contiguity spatial-weighting matrix.

. /* Create a spatial contiguity matrix */
. spmatrix create contiguity W

Now that I have my data and my spatial-weighting matrix, I can estimate the model parameters.

. /* Estimate SAR model parameters */
. spregress hrate unemployment, dvarlag(W) gs2sls
  (254 observations)
  (254 observations (places) used)
  (weighting matrix defines 254 places)

Spatial autoregressive model                    Number of obs     =        254
GS2SLS estimates                                Wald chi2(2)      =      14.23
                                                Prob > chi2       =     0.0008
                                                Pseudo R2         =     0.0424

------------------------------------------------------------------------------
       hrate |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
hrate        |
unemployment |   .4584241    .152503     3.01   0.003     .1595237    .7573245
       _cons |   2.720913   1.653105     1.65   0.100    -.5191143    5.960939
-------------+----------------------------------------------------------------
W            |
       hrate |   .3414964   .1914865     1.78   0.075    -.0338103    .7168031
------------------------------------------------------------------------------
Wald test of spatial terms:          chi2(1) = 3.18       Prob > chi2 = 0.0745

Spatial spillover

Now we are ready to answer the second question. Based on our estimation results from spregress, we can proceed in three steps.

  1. Predict the homicide rate using original data.
  2. Change Dallas’s unemployment rate to 10% and predict the homicide rate again.
  3. Compute the difference between two predictions and map it.
. preserve /* save data temporarily */

. /* Step 1: predict homicide rate using original data */
. predict y0
(option rform assumed; reduced-form mean)

. /* Step 2: change Dallas unemployment rate to 10%, and predict again*/
. replace unemployment = 10 if cname == "Dallas"
(1 real change made)
. predict y1
(option rform assumed; reduced-form mean)

. /* Step 3: Compute the prediction difference and map it*/
. generate double y_diff = y1 - y0
. grmap y_diff, title("Global spillover")
. restore /* return to original data */

The above graph shows that a change in the unemployment rate in Dallas changes the homicide rates in the counties that are near to Dallas, in addition to the homicide rate in Dallas. The change in Dallas spills over to the nearby counties, and the effect is known as a spillover effect.  

连享会计量方法专题……

SAR model and spatial spillover

In this section, I show why a SAR model generates a spillover effect. In the process, I provide a formula for this effect that I use to create the animated graph.
The matrix form for a SAR model is

Solving for y yields

The mean value of y given a value of X is known as the the expectation of y conditional on X. Because ϵ is independent of X , the expectation of y conditional on X is

Note that this conditional expectation specifies the mean for each county in Texas because y is a vector.

We use this equation to define the effect of going from one set of values for X to another set. In the case at hand, I let X0 contain the covariate values in the observed data and let X1 contain the same values except that the unemployment rate in Dallas has been set to 10%. With this notation, I see that going from X0 to X1 causes the mean homicide rates for each county in Texas to change by

where ΔX=X1X0.

I now show that a technical condition assumed in SAR models produces an expression for the animated graph. SAR models are widely used because they satisfy a stability condition. Intuitively, this stability condition says that the inverse matrix (IλW)1 can be written as a sum of terms that decrease in size exponentially fast. This condition is that

Plugging the formula from (2) into the effect in (1) yields

which is the expression for the effect that I use to generate the animated graph.

Each term in (3) has some intuition, which is most easily presented in terms of my example. The first term (ΔXβ) is the initial effect of the change, and it affects only the homicide rate in Dallas. The second term (λWΔXβ) is the effect of the change on the outcome in those places that are neighbors of Dallas. The third term (λ2W2ΔXβ) is the effect of the change on the outcome in those places that are neighbors of neighbors of Dallas. The intuition continues in the pattern for the remaining terms.

Create animated graphs for spillover effects

I now describe how I generate the animated graph. Each graph plots the change using a subset of the terms in (3). The first graph plots the change computed from the first term only. The second graph plots the change computed from the first and second terms only. The third graph plots the change computed from the first three terms only. And so on.

The first four steps of the code do the following.

  1. It computes and plots ΔXβ.
  2. It computes and plots ΔXβ+λWΔXβ.
  3. It compute and plots ΔXβ+λWΔXβ+λ2W2ΔXβ.
  4. It computes and plots ΔXβ+λWΔXβ+λ2W2ΔXβ+λ3W3ΔXβ.

Steps 5 through 20 perform the analogous operations.
Finally, combine graphs from step 1 to step 20, and create an animated graph.
Here is the code that implements this process.

  1 /* get estimate of spatial lag parameter lambda */
  2 local lambda = _b[W:hrate]
  3
  4 /* xb based on original data */
  5 predict xb0, xb
  6
  7 /* xb based on modified data */
  8 replace unemployment = 10 if cname == "Dallas"
  9 predict xb1, xb
 10
 11 /* compute the outcome change in the first step */
 12 generate dy = xb1 - xb0
 13 format dy %9.2f
 14
 15 /* Initialize Wy, lamWy, */
 16 generate Wy = dy
 17 generate lamWy = dy
 18
 19 /* map the outcome change in step 1 */
 20 grmap dy
 21 graph export dy_0.png, replace
 22 local input dy_0.png
 23
 24 /* compute the outcome change from step 2 to 11 */
 25 forvalues p=1/20 {
 26         spgenerate tmp = W*Wy
 27         replace lamWy = `lambda'^`p'*tmp
 28         replace Wy = tmp
 29         replace dy = dy + lamWy
 30         grmap dy
 31         graph export dy_`p'.png, replace
 32         local input `input' dy_`p'.png
 33         drop tmp
 34 }
 35
 36 /* convert graphs into a animated graph */
 37 shell convert -delay 150 -loop 0 `input' glsp.gif
 38
 39 /* delete the generated pgn file */
 40 shell rm -fR *.png

This code uses the ereturn results produced by spregress above and its corresponding predictcommand.

  • Line 2 puts the estimate of λ in the local macro lambda.
  • Lines 5, 7, 8, and 9 compute Xβ for X0 and X1 and store them in xb0 and xb1, respectively.
  • Line 12 computes the first term (ΔXβ) and stores it in dy.
  • Lines 16 and 17 store the initial values for Wpy and λpWpy, when p=0.
  • Lines 20–22 produce the first plot in the animated graph. The local macro input will contain all the plots used to create the animated graph when the code finishes.
  • Lines 25–34 compute the terms and create the plots for the remaining terms. Line 26 usesspgenerate to compute Wpy. Line 27–33 perform operations analogous to those of dy.
  • In Line 37, I use a Linux tool “convert” to combine the graphs to produce an animated graph. On Windows, I can use software such as FFmpeg and Camtasia. For more details, see How to create animated graphics using Stata by Chuck Huber.
  • Line 40 deletes all the unnecessary .png files.

Here is the animated graph created by this code.

Done and undone

In this post, I discussed spillover effects and why SAR models produce them in the context of an example using the counties in Texas. I also showed how the effects can be computed as an accumulated sum. I used the accumulated sum to create an animated graph that illustrates how the effects spill over in the counties in Texas.

相关课程

连享会-直播课 上线了!
http://lianxh.duanshu.com

免费公开课:


课程一览

支持回看,所有课程可以随时购买观看。

专题 嘉宾 直播/回看视频
最新专题 DSGE, 因果推断, 空间计量等
Stata数据清洗 游万海 直播, 2 小时,已上线
研究设计 连玉君 我的特斯拉-实证研究设计-幻灯片-
面板模型 连玉君 动态面板模型-幻灯片-
面板模型 连玉君 直击面板数据模型 [免费公开课,2小时]

Note: 部分课程的资料,PPT 等可以前往 连享会-直播课 主页查看,下载。


关于我们

  • Stata连享会 由中山大学连玉君老师团队创办,定期分享实证分析经验。直播间 有很多视频课程,可以随时观看。
  • 连享会-主页知乎专栏,300+ 推文,实证分析不再抓狂。
  • 公众号推文分类: 计量专题 | 分类推文 | 资源工具。推文分成 内生性 | 空间计量 | 时序面板 | 结果输出 | 交乘调节 五类,主流方法介绍一目了然:DID, RDD, IV, GMM, FE, Probit 等。
  • 公众号关键词搜索/回复 功能已经上线。大家可以在公众号左下角点击键盘图标,输入简要关键词,以便快速呈现历史推文,获取工具软件和数据下载。常见关键词:课程, 直播, 视频, 客服, 模型设定, 研究设计, stata, plus, 绘图, 编程, 面板, 论文重现, 可视化, RDD, DID, PSM, 合成控制法

连享会主页  lianxh.cn
连享会主页 lianxh.cn

连享会小程序:扫一扫,看推文,看视频……

扫码加入连享会微信群,提问交流更方便

✏ 连享会学习群-常见问题解答汇总:
https://gitee.com/arlionn/WD

参考资料

[1]

Posts by Di Liu, Senior Econometrician: https://blog.stata.com/author/dliu/