CaseStudyReport
Carlos Mercado
November 7, 2017
Intro
“William Phillips, a New Zealand born economist, wrote a paper in 1958 titled The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957, which was published in the quarterly journal Economica. In the paper Phillips describes how he observed an inverse relationship between inflation and unemployment in the British economy over the period examined. This relationship became known as the Phillip’s Curve.”
First, does this relationship hold in the United States?
Second, is there more to the relationship than what William Phillips found?
Fast Answer
- William Phillips, having written his paper in 1958, was years ahead of his time. The Phillips Curve almost perfectly describes the US in the 1960s, but has lost its predictive abilities since. Overall, a statistically significant relationship has existed between the two. But, I found that significant problems arise when attempting to prove the relationship in the short-run.
Data Cleaning and Thought Process
- Data uploaded to github
fileUrl <- "https://github.com/RafaJones/CaseStudyData/raw/master/FI%20Case%20Data%20(1).csv"
download.file(fileUrl, destfile = "./CpiUnemployment.csv")
CpiUnemployment <- read.csv("CpiUnemployment.csv", stringsAsFactors = FALSE)- Data Cleaning
- Two sided Approach
- Long Run Analysis
- Short Run Perspective & Problems
Data Cleaning
- Are there any misspellings or mislabelings?
- Any missing Data?
- Any impossible data?
UniqueVals <- sapply(CpiUnemployment,unique) #Check the first three columns
UniqueVals[1:3]## $Data
## [1] "Consumer Price Index" "Unemployment Level" "Civilian Labor Force"
## [4] "CPI"
##
## $Year
## [1] 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960
## [15] 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974
## [29] 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988
## [43] 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
## [57] 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016
## [71] 2017
##
## $Month
## [1] 1 2 3 4 5 6 7 8 9 10 11 12#CPI and Consumer Price Index should be the same
#correct number of years - CPI and Consumer Price Index are the same
- The years and months are correct
Data Cleaning
- Is it really true that no data is missing? Originally, 0% of each column is coded as missing (NA)
avgNA <- function(x) mean(is.na(x)) #this function identifies proportion of NA
sapply(CpiUnemployment,avgNA)## Data Year Month Value
## 0 0 0 0#No current NAs, as expected from UniqueVals, of course the Value column
#may still have impossible data, that will be checked later
table(as.data.frame(CpiUnemployment[,1])) #how many of each Data is there? ##
## Civilian Labor Force Consumer Price Index CPI
## 276 804 38
## Unemployment Level
## 830- Yet there are an unequal number of “Data” entries
- Not Equal = The missing data was hidden!
Data Cleaning
- Reshape the data into proper format: every row an observation and every column an element of that single observation.
- Combine “Consumer Price Index” and “CPI” into one name
- Expect Civilian Labor Force data to be missing
#First change "Consumer Price Index" to "CPI" to match the data
CpiUnemployment$Data <- sub("Consumer Price Index","CPI",CpiUnemployment$Data)
#Reshape the data into tidy format
library(tidyr)
CPI <- data.frame(CpiUnemployment)
CPI <- spread(CPI,Data,Value)
#Note this data stops at February 2017
datescheck <- table(CPI[,1:2]) #confirm that each year has each month
#Changing month and year to a yearmon (date) object instead of strings
suppressMessages(library(zoo))
CPI$Date <- as.yearmon(paste(CPI$Year,CPI$Month),"%Y %m")
CPI[9:15,]## Year Month Civilian Labor Force CPI Unemployment Level Date
## 9 1947 9 NA 22.84 NA Sep 1947
## 10 1947 10 NA 22.91 NA Oct 1947
## 11 1947 11 NA 23.06 NA Nov 1947
## 12 1947 12 NA 23.41 NA Dec 1947
## 13 1948 1 60230 23.68 2034 Jan 1948
## 14 1948 2 NA 23.67 2328 Feb 1948
## 15 1948 3 NA 23.50 2399 Mar 1948- A sneak peak at a few early rows
- Besides 1947, most data is available.
- The data has been reformmated, adding NA for Civilian Labor Force data
- Every year has every month except 2017 - the data stops at February 2017
Data Cleaning
- A more accurate measure of missing data
#Revisit the percentage of NAs
sapply(CPI, avgNA)## Year Month Civilian Labor Force
## 0.00000000 0.00000000 0.67220903
## CPI Unemployment Level Date
## 0.00000000 0.01425178 0.00000000While Unemployment Levels are available for every year since 1948 and CPI available totally, labor force levels are only recorded for about 1/3rd of the time. Essentially, every third month since 1948.
To get a full picture of the Phillips Curve I want to approach from two angles: - First - using quarterly data (i.e. the 1/3rd of data with all elements) I want to check for a general relationship between inflation (% change in CPI) and unemployment rate (level divided by labor force) in the Long Run.
- Second, imputing missing labor force data from nearby data, I want to test the hypothesis that small changes in prices (% change monthly) correlate with small changes in unemployment (% change in unemployment level) in the Short Run.
The Phillips Curve in the US Long Run
- Quarterly Data 1948 - 2017
- Inflation - % change in CPI
- Unemployment Rate - % of labor force, unemployed
#relevant functions
percentchange <- function(xvector){
output <- rep(0,length(xvector)) #create an empty vector
for(i in 2:length(xvector)){ #the first has 0 inflation to itself
output[i] <- (xvector[i]-xvector[i-1])/xvector[i-1]
} #for each element after the first
return(output)
}
#quarterlydata since 1948
CPIquarterly48 <- CPI[complete.cases(CPI),]
suppressMessages(library(dplyr))
#create inflation column
CPIquarterly48 <- mutate(CPIquarterly48, inflation.percent = round(100 * percentchange(CPI),2))
#create unemployment rate column
CPIquarterly48 <- mutate(CPIquarterly48, Unem.Rate = round(100 * `Unemployment Level` / `Civilian Labor Force`,2))
#simple plot
suppressMessages(library(ggplot2))
g <- ggplot(data = CPIquarterly48, aes(Unem.Rate,inflation.percent))
gg <- g + geom_point() + geom_point(aes(colour=Year))
ggg <- gg + geom_smooth(method = "lm") The Phillips Cuve in the Long Run
-Generally speaking, inflation and unemployment rates aren’t related in the long-run. This fits the theory developed in the 1970’s that a “natural” rate of unemployment exists
ggg
The Phillips Curve in the Long Run
- Adjusting for Phillips Curve Assumptions
#select only data that has positive inflation
CPIquarterly48p <- CPIquarterly48[CPIquarterly48$inflation.percent > 0,]
#create simple graph with two smoothing lines
pg <- ggplot(data = CPIquarterly48p, aes(Unem.Rate,inflation.percent))
pgg <- pg + geom_point() + geom_point(aes(colour=Year))
pggg <- pgg + geom_smooth(method = "loess", se= FALSE) +
geom_smooth(method = "lm",se = FALSE, colour = "red") +
labs(title = "Positive Inflation Only",x="Unemployment(%)",
y="Inflation(%)", caption = "Red = Linear Model, Blue = Loess")
suppressMessages(pggg)
- NAIRU holds in the long run, no noticeable relationship.
The Phillips Curve in the US Short Run
Long run as envelopes of the short run, i.e. Inter-decade phillips curves
Splitting the data by decades to see if any particular time period shows a similar relationship
The 1960s look exactly like a Phillips Curve!
decadelabel <- function(year){
if(year < 1950) label <- "40s"
else if(year < 1960) label <- "50s"
else if(year < 1970)label <- "60s"
else if(year < 1980) label <- "70s"
else if(year < 1990) label <- "80s"
else if(year < 2000) label <- "90s"
else if(year < 2010) label <-"2000's"
else if(year < 2020) label <- "2010's"
return(label)
}
suppressMessages(library(dplyr))
CPIquarterly48pd <- mutate(CPIquarterly48p, Decade = sapply(Year,decadelabel))
dg <- ggplot(data = CPIquarterly48pd,
aes(Unem.Rate,inflation.percent, group = Decade)) +
facet_wrap(~Decade)
dgg <- dg + geom_point() +geom_point(aes(color = Decade)) +
geom_smooth(method = "loess", se = FALSE, aes(color = Decade))
dgg <- dgg +
labs(title = "Decade Phillips Curves", y = "Inflation (%)",
x = "Unemployment Rate (%)", subtitle = "Based on Quarterly data",
caption = "Note: Missing quarters had deflation")
suppressWarnings(suppressMessages(dgg))
The Phillips Curve in the Short Run - Allowing Deflation
- Keeping the decades grouping, can we identify the correlation between changes in inflation and unemployment rates. That is, second derivatives of prices and unemployment.
To do this, I want to do three things:
I want to impute missing civilian labor force values by reverting to the values of the previous quarter. These errors will be small, but worth it, for assessing short term changes in unemployment. Thus, we get Monthly instead of Quarterly data.
Calculate the changes in inflation rate and unemployment rate
Test the significance of their correlation by decade (short run) and in aggregate (long run)
#original dataset, taking off 1947 and after correcting names
CPI48 <- CPI[-c(1:12),]
#backup function for the missing Civilian Labor Force
backupreplace <- function(input){ #this function
for(i in 1:length(input)){ #goes through each element in a vector
j <- i
while(is.na(input[i])) #and while the element is NA
input[i] <- input[j-1]
j <- j - 1 #it makes it match the value behind it
} #until it is no longer an NA
return(input)
}
CPI48$`Civilian Labor Force` <- backupreplace(CPI48$`Civilian Labor Force`)
#impute the labor force as the most previous available value
library(dplyr)
CPI48 <- mutate(CPI48, Unem.Rate = round(100 * `Unemployment Level` / `Civilian Labor Force`,2)) #make the imputed unemployment rate in % terms
CPI48 <- mutate(CPI48, inflation = round(100 * percentchange(CPI),2))
#make the inflation rate in % terms
suppressMessages(library(dplyr))
CPI48d <- mutate(CPI48, Decade = sapply(Year,decadelabel))
diffchange <- function(xvector){
output <- rep(0,length(xvector)) #create an empty vector
for(i in 2:length(xvector)){ #the first has 0 inflation to itself
output[i] <- (xvector[i]-xvector[i-1])
} #for each element after the first
return(output)
} #just takes the difference between an element and its previous element.
CPI48dc <- mutate(CPI48d, dinflation = diffchange(inflation), dUnem.Rate = diffchange(Unem.Rate)) #add new rows for changes in each element
d48grouped <- group_by(CPI48dc, Decade) #group it by decade
head(d48grouped[,7:11]) #preview ## # A tibble: 6 x 5
## # Groups: Decade [1]
## Unem.Rate inflation Decade dinflation dUnem.Rate
## <dbl> <dbl> <chr> <dbl> <dbl>
## 1 3.38 0.00 40s 0.00 0.00
## 2 3.87 -0.04 40s -0.04 0.49
## 3 3.98 -0.72 40s -0.68 0.11
## 4 3.94 1.36 40s 2.08 -0.04
## 5 3.50 0.80 40s -0.56 -0.44
## 6 3.66 0.58 40s -0.22 0.16Time Series Plots
tsCPI48dc <- ts(CPI48dc[,-c(1,2,9)]) #note: Date is already a date object
#removing the Year and Month columns, also removing the factor Decade
#renaming to improve visual
colnames(tsCPI48dc) <- c("Civilian Labor Force","CPI","Unemployment #",
"Date","Unemp.%","Inflation %",
"Change.I%","Change.U%"
)
plot.ts(tsCPI48dc[,-c(1:4)], main = "Relevant Time Plots",xlab = "Months since 1948" ) 
#removing a few plot for brevity and a little
#obviousnessA Short Run Analysis - Allowing Deflation
- Does a significant relationship exist between small changes in the inflation rate and small changes in the unemployment rate within a decade?
summarise(d48grouped, `Correlation of small changes` = round(cor(dUnem.Rate,dinflation),2),
Significance = round(cor.test(dUnem.Rate,dinflation)$p.value,2))## # A tibble: 8 x 3
## Decade `Correlation of small changes` Significance
## <chr> <dbl> <dbl>
## 1 2000's 0.05 0.59
## 2 2010's -0.06 0.59
## 3 40s -0.27 0.21
## 4 50s -0.07 0.42
## 5 60s 0.07 0.44
## 6 70s -0.03 0.71
## 7 80s -0.26 0.00
## 8 90s 0.05 0.56Generally, no. The 1980s, with a few periods of abnormally high unemployment being the exception. Small changes in inflation were well correlated at high levels of inflation and unemployment.
A short Run Analysis - Allowing Deflation
- Does a significant relationship exist for the inflation rate and unemployment rate themselves within a decade?
summarise(d48grouped, `Correlation as a whole` = round(cor(Unem.Rate,inflation),2),
Significance = round(cor.test(Unem.Rate,inflation)$p.value,2))## # A tibble: 8 x 3
## Decade `Correlation as a whole` Significance
## <chr> <dbl> <dbl>
## 1 2000's -0.13 0.16
## 2 2010's 0.11 0.33
## 3 40s -0.28 0.19
## 4 50s -0.12 0.19
## 5 60s -0.56 0.00
## 6 70s -0.14 0.13
## 7 80s -0.10 0.28
## 8 90s 0.12 0.21Generally, no. This relates back to the plot of positive-only inflation. Besides the 1960s, exogenous influences prevent a direct relationship from being predictable.
A Final Word
- Using all the available data, small increases in the inflation rate are significantly correlated with small decreases in the unemployment rate. Breaking it apart by decade, although still 120 data points of monthly data each, reduces our ability to identify statistical significance.
cor.test(CPI48dc$dinflation,CPI48dc$dUnem.Rate) #correlation test ##
## Pearson's product-moment correlation
##
## data: CPI48dc$dinflation and CPI48dc$dUnem.Rate
## t = -2.1894, df = 828, p-value = 0.02885
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.143177512 -0.007858682
## sample estimates:
## cor
## -0.0758674Conclusions
- Does the relationship between inflation and unemployment hold in the US?
Yes, a relationship between the two exists. In the long run, small changes in the inflation rate are related to small changes in the unemployment rate.
Uniquely, the relationship was most pronounced in the 1960’s. Afterwards, the same concerns that plagued the theory in the US (the existence of stagflation in the 70s along with high unemployment) muddle the relationship.
Within any one decade, the Phillips Curve hasn’t adequately described the general relationship between the two. This may be due to exogenous influences unique to the decade (e.g. stagflation or changes in government spending).
A change in the what could best be described as the “natural rate” of inflation is related to a change in the “natural rate” unemployment.