The Wooldridge texts are part of most graduate level econometric studies in the US. They are:
Introductory Econometrics: A Modern Approach, 6th Edition
Econometric Analysis of Cross Section and Panel Data, 2nd Edition
I would start with the first, unless you have a lot of formal training in statistics.
Hope this helps,
Justin
econometrics macroeconomics statistics r
UPDATED EXAMPLES: from INTRODUCTORY ECONOMETRICS by Jeffrey M. Wooldridge
While my suggested texts about Econometrics were downvoted, below is proof of how the text book referenced illustrates how Economists can use statistical methods to identify and measure leading, lagging, and coincident relationships among economic indicators.
The first examples are from chapter 16 on Simultaneous Equations. Examples 16.4 and 16.6 illustrate how to use Instrumental-variable regression by two-stage least squares for identifying and measuring coincident economic indicators. In this case, how annual inflation drops by about 1/3 of a percent for every 1 percent increase in the import share of GDP.
The second is an example 18.8 of the text and illustrate how autoregresive $AR(1)$ and vector autoregressive $VAR(1)$ time series methods can be used to identify and measure, intertemporal relationships between economic indicators, in an attempt to forecast the future unemployment rate using today's unemployment and inflation rates.
Example 16.4: INFLATION AND OPENNESS
"Romer (1993) proposes theoretical models of inflation that imply that more "open" countries should have lower inflation rates. His empirical analysis explains average annual inflation rates (since 1973) in terms of the average share of imports in gross domestic product since 1973 - which is his measure of openness. In addition to estimating
the key equation by OLS, he uses instrumental variables. While Romer does not specify
both equations in a simultaneous system, he has in mind a two-equation system:"
$$inf = \beta_{10} + \alpha_1open + \beta_{11}log(pcinc) + \mu_1$$
$$open = \beta_{20} + \alpha_2inf + \beta_{21}log(pcinc) + \beta_{22}log(land) + \mu_2$$
"where $pcinc$ is 1980 per capita income, in U.S. dollars, assumed to be exogenous, and
$land$ is the land area of the country in square miles, also assumed to be exogenous. The first equation is the one of interest, with the hypothesis that $\alpha < 0$. More open
economies have lower inflation rates."
"The second equation reflects the fact that the degree of openness might depend on the average inflation rate, as well as other factors. The variable $log(pcinc)$ appears in both equations, but $log(land)$ is assumed to appear only in the second equation.
The idea is that, ceteris paribus, a smaller country is likely to be more open, so $\beta_{22} < 0$."
"Using the identification rule that was stated earlier, the first equation is identified, provided $\beta_{22} \ne 0$. The second equation is $not$ identified because it contains both exogenous variables. Be we are interested in the first equation.
Example 16.6: INFLATION AND OPENNESS
"Before we estimate the first equation in 16.4 using the data in $openness$, we check to see whether $open$ has sufficient partial correlation with the proposed $IV$, $log(land)$. The reduced form regression is:"
$$\widehat{open} = \beta_0 + \beta_{1}log(pcinc) + \beta_{2}log(land)$$
open_model <-lm(open ~ lpcinc + lland, data = openness)
summary(open_model)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 117.0845 15.8483 7.388 2.97e-11
lpcinc 0.5465 1.4932 0.366 0.715
lland -7.5671 0.8142 -9.294 1.51e-15
"The $t$ statistic on $log(land)$ is over nine in absolute value, which verifies Romer's assertion that smaller countries are more open. The fact that $log(pcinc)$ is so insignificant in this regression is irrelevant."
"Estimating the first equation using $log(land)$ as an $IV$ for $open$ gives:"
$$\widehat{inf} = \beta_0 + \beta_{1}open + \beta_{2}log(pcinc)$$
library(AER)
inflation_IV <- ivreg(inf ~ open + lpcinc | lpcinc + lland, data = openness)
summary(inflation_IV)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 26.8993 15.4012 1.747 0.0835 .
open -0.3375 0.1441 -2.342 0.0210
lpcinc 0.3758 2.0151 0.187 0.8524
"The coefficient on open is statistically significant at about the 1% level against a one sided alternative of $\alpha_1 < 0$. The effect is economically important as well: for every percentage point increase in the import share of GDP, annual inflation is about 1/3 of a percentage point lower. For comparison, the OLS estimate is -0.215, $se = 0.095$."
Example 18.8: FORECASTING THE U.S. UNEMPLOYMENT RATE
"We use the $PHILLIPS$ DATA, but only for the years 1948 through 1996, to forecast
the U.S. civilian unemployment rate for 1997. We use two models. The first is a simple
AR(1) model for $unem$:"
$$\widehat{unemp_t} = \beta_0 + \beta_1unem_{t-1}$$
"In a second model, we add inflation with a lag of one year:"
$$\widehat{unemp_t} = \beta_0 + \beta_1unem_{t-1} + \beta_2inf_{t-1}$$
library(dynlm)
phillips <- ts(phillips, start = 1948)
unem_AR1 <- dynlm(unem ~ unem_1, data = phillips, end = 1996)
unem_inf_VAR1 <- dynlm(unem ~ unem_1 + inf_1, data = phillips, end = 1996)
stargazer(unem_AR1, unem_inf_VAR1, keep.stat=c("n","adj.rsq","ser")
=================================================================
Dependent variable:
---------------------------------------------
unem
(1) (2)
-----------------------------------------------------------------
unem_1 0.732 0.647
(0.097) (0.084)
inf_1 0.184
(0.041)
Constant 1.572 1.304
(0.577) (0.490)
-----------------------------------------------------------------
Observations 48 48
R2 0.554 0.691
Adjusted R2 0.544 0.677
Residual Std. Error 1.049 (df = 46) 0.883 (df = 45)
F Statistic 57.132 (df = 1; 46) 50.219 (df = 2; 45)
"The lagged inflation rate is very significant in the second model $(t \approx 4.5)$, and the adjusted R-squared much higher than that from the first. Nevertheless, this does not necessarily mean that the second equation will produce a better forecast for 1997. All we can say so far is that, using the data up through 1996, a lag of inflation helps to explain variation in the unemployment rate."
"To obtain the forecasts for 1997, we need to know $unemployment$ and $inflation$ in 1996. These are 5.4 and 3.0, respectively. Therefore, the forecast of $unem_{1997}$ from the first equation is $1.572 + .732(5.4)$, or about $5.52$. The forecast from the second equation is $1.304 + 0.647(5.4) + 0.184(3.0)$, or about $5.35$. The actual civilian unemployment rate for 1997 was $4.9$, so both equations over-predict the actual rate. The second equation does provide a somewhat better forecast."
