Time-series study of the proportion of food in Personal consumption expenditure

Econ 385

I think that to model time series (1967-2016) differences in the proportion of Food Expenditure in personal consumption expenditure based on Personal Income, Inflation consumer prices, and the proportion of Health Care Expenditure in Personal Consumption Expenditure. So I think that the dependent variable is the Proportion of Food Expenditure in personal consumption expenditure (PFIA), and explanatory variable is Personal Income (PI), and the proportion of Health Care Expenditure in Personal Consumption Expenditure (PHCIA). I found the data of the personal consumption expenditure: food, and the personal consumption expenditure: health care, calculate their proportion in the personal consumption expenditure: All term. Meanwhile I assume that the proportion of Food Expenditure in personal consumption expenditure will decrease as personal income increase, because people will pay more money to other parts such as Health Care rather than pay to Food, when their Income increase, and we can found it form the scatter plot(1), the proportion health care expenditure in personal consumption expenditure has a same relationship and tendency, like scatter plot(2)

(1)

 

 

(2)

 

 

The MEANS Procedure

Variable Label N Mean Std Dev Minimum Maximum
PFIA

PI

PHCIA

PFIA

PI

PHCIA

49

49

49

0.1073261

6151.41

0.1237509

0.0307650

4544.49

0.0320319

0.0733248

665.7000000

0.0628695

0.1627907

15458.50

0.1684346

proc means;

var PFIA PI PHCIA;

run;

From this MEANS Procedure, the mean of PFIA is 0.173, the mean of PI is 6154.41, and the mean of PHCIA is 0.124, meanwhile we can found that as the PFIA decrease, the personal income and PHCIA will be increase.

 The Relationship PFIA with PI and PHCIA

The REG Procedure

Model: MODEL1

Dependent Variable: PFIA PFIA

Number of Observations Read 50
Number of Observations Used 49
Number of Observations with Missing Values 1

 

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 2 0.04317 0.02159 439.86 <.0001
Error 46 0.00226 0.00004908
Corrected Total 48 0.04543

 

Root MSE 0.00701 R-Square 0.9503
Dependent Mean 0.10733 Adj R-Sq 0.9481
Coeff Var 6.52731  

 

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept 1 0.22309 0.00700 31.86 <.0001
PI PI 1 -1.06236E-8 5.713825E-7 -0.02 0.9852
PHCIA PHCIA 1 -0.93490 0.08106 -11.53 <.0001

proc reg;

title’The Relationship PFIA with PI and PHCIA’;

model PFIA=PI PHCIA;

run;

I used SAS to perform an Ordinary Least Squares analysis of my data set. Below is the interpretation of my regression results. For this sample an increase one percentage in the personal income was associated with an additional -1.06236E-8 dollars in the proportion food expenditure of personal consumption expenditure, holding the proportion of health care expenditure in personal consumption expenditure constant. The coefficient on the personal income is not statistically significant (p=0.9852). And the proportion of health care expenditure in personal consumption expenditure is highly statistically significant (p=<0.0001), The regression as a whole fit reasonably well (R2=0.9503, adjusted R2=0.9481) and was highly statistically significant (F=439.86, P<0.0001).

 

Firstly I suspect that my regression has problem of multicollinearity, I want to run PROC CORR to detect the multicollinearity, and I can found that the correlation coefficients is bigger than 0.8(0.92106) in explanatory variable-personal income and explanatory variable-the proportion of health care expenditure in the personal consumption expenditure. So I think that the regression has problem of multicollinearity.

The Relationship PFIA with PI and PHCIA

The CORR Procedure

3 Variables: PFIA PI PHCIA

 

Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum Label
PFIA 49 0.10733 0.03077 5.25898 0.07332 0.16279 PFIA
PI 49 6151 4544 301419 665.70000 15459 PI
PHCIA 49 0.12375 0.03203 6.06379 0.06287 0.16843 PHCIA

 

Pearson Correlation Coefficients, N = 49
Prob > |r| under H0: Rho=0
  PFIA PI PHCIA
PFIA

PFIA

1.00000

 

-0.89813

<.0001

-0.97484

<.0001

PI

PI

-0.89813

<.0001

1.00000

 

0.92106

<.0001

PHCIA

PHCIA

-0.97484

<.0001

0.92106

<.0001

1.00000

 

 

proc corr;

var PFIA PI PHCIA;

run;

I will make sure again for multicollinearity, so I want to make variance inflation factor test. And I still found that these two explanatory VIF is so higher, bigger than 5, the PI is 6.59445 and the PHCIA is 6.59445. I want to cure it.

The Relationship PFIA with PI and PHCIA

The REG Procedure

Model: MODEL1

Dependent Variable: PFIA PFIA

Number of Observations Read 50
Number of Observations Used 49
Number of Observations with Missing Values 1

 

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 2 0.04317 0.02159 439.86 <.0001
Error 46 0.00226 0.00004908
Corrected Total 48 0.04543

 

Root MSE 0.00701 R-Square 0.9503
Dependent Mean 0.10733 Adj R-Sq 0.9481
Coeff Var 6.52731  

 

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t| Variance
Inflation
Intercept Intercept 1 0.22309 0.00700 31.86 <.0001 0
PI PI 1 -1.06236E-8 5.713825E-7 -0.02 0.9852 6.59455
PHCIA PHCIA 1 -0.93490 0.08106 -11.53 <.0001 6.59455

 

proc reg;

model PFIA = PI ICP PHCIA/ VIF;

RUN;

I want to cure it, drop one of the multicollinearity variable- the proportion of health care. And we can found that an increase of one percentage point in the personal income was associated with an additional -0.00000608dollars in the proportion of food expenditure in the personal consumption expenditure, and the coefficient on the personal income was highly statistically significant (p<0.0001). The result is my estimation, because from the Engel Curve, we can know that as the personal income increase, the proportion of food expenditure in the personal consumption expenditure will decrease, because people will pay more attention to others’ part.

The Relationship PFIA with PI and PHCIA

The REG Procedure

Model: MODEL1

Dependent Variable: PFIA PFIA

Number of Observations Read 50
Number of Observations Used 49
Number of Observations with Missing Values 1

 

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 1 0.03665 0.03665 196.06 <.0001
Error 47 0.00879 0.00018692
Corrected Total 48 0.04543

 

Root MSE 0.01367 R-Square 0.8066
Dependent Mean 0.10733 Adj R-Sq 0.8025
Coeff Var 12.73848  

 

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept 1 0.14473 0.00331 43.74 <.0001
PI PI 1 -0.00000608 4.34228E-7 -14.00 <.0001

proc reg;

title’The Relationship PFIA with PI;

model PFIA=PI;

run;

 

 

 

Because it is a time series project, so I estimate it is an autocorrelation, and take it to DW test. With three explanatory variables and 49 observations, and I get Durbin Watson d statistic d, 5%significance level, dLis 1.5, and dU is 1.59, so form the DW d less than dL 1.50, I will reject H0, no positive autocorrelation, so it is an autocorrelation.

The Relationship PFIA with PI and PHCIA

The REG Procedure

Model: MODEL1

Dependent Variable: PFIA PFIA

Durbin-Watson D 0.276
Number of Observations 49
1st Order Autocorrelation 0.848

 

proc reg;

model PFIA=PI PHCIA/DW;

run;

 

 

Fix the autocorrelation

Relationship PFTA with PI AND PHCIA

The AUTOREG Procedure

Dependent Variable PFIA
  PFIA

Relationship PFTA with DSPIC CPIF and PHCIA

The AUTOREG Procedure

Ordinary Least Squares Estimates
SSE 0.00225755 DFE 46
MSE 0.0000491 Root MSE 0.00701
SBC -338.54805 AIC -344.22351
MAE 0.00572767 AICC -343.69018
MAPE 6.22501397 HQC -342.07025
Durbin-Watson 0.2765 Total R-Square 0.9503

 

Durbin-Watson Statistics
Order DW Pr < DW Pr > DW
1 0.2765 <.0001 1.0000

NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.

Parameter Estimates
Variable DF Estimate Standard
Error
t Value Approx
Pr > |t|
Variable Label
Intercept 1 0.2231 0.007003 31.86 <.0001
PI 1 -1.062E-8 5.7138E-7 -0.02 0.9852 PI
PHCIA 1 -0.9349 0.0811 -11.53 <.0001 PHCIA

 

Estimates of Autocorrelations
Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
0 0.000046 1.000000 |                    |********************|
1 0.000039 0.847528 |                    |*****************   |

 

Preliminary MSE 0.000013

 

Estimates of Autoregressive Parameters
Lag Coefficient Standard
Error
t Value
1 -0.847528 0.079120 -10.71

Relationship PFTA with DSPIC CPIF and PHCIA

The AUTOREG Procedure

Yule-Walker Estimates
SSE 0.00043337 DFE 45
MSE 9.63056E-6 Root MSE 0.00310
SBC -414.26047 AIC -421.82775
MAE 0.00248173 AICC -420.91866
MAPE 2.40552271 HQC -418.95673
Durbin-Watson 0.9490 Transformed Regression R-Square 0.7794
  Total R-Square 0.9905

 

Durbin-Watson Statistics
Order DW Pr < DW Pr > DW
1 0.9490 <.0001 1.0000

NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for testing negative autocorrelation.

Parameter Estimates
Variable DF Estimate Standard
Error
t Value Approx
Pr > |t|
Variable Label
Intercept 1 0.1867 0.009900 18.86 <.0001
PI 1 -2.519E-6 7.998E-7 -3.15 0.0029 PI
PHCIA 1 -0.4990 0.1085 -4.60 <.0001 PHCIA

 

proc autoreg;

title’Relationship PFTA with DSPIC CPIF and PHCIA’;

MODEL PFIA=PI ICP PHCIA/NLAG=1 DWPROB;

RUN;

Conclusion:

I estimate that as the personal income increase, the proportion of food expenditure in the personal consumption expenditure. Because in the Engle curve when personal income increase, people will pay more money to others part rather than food, maybe people will pay more attention to health care and sport activities. However, my project still has some limitation, for example, I should add more explanatory, because I think that a lot of factor will affect the change of the proportion of food expenditure in the personal consumption expenditure. On the other hand, I should add different state data to compare, it can help me analysis different state and different family will have different results in the project.

The relationship PFIA with PI and PHCIA