Transcript Financial Analysis, Planning and Forecasting Theory and

Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 2
Accounting Information, Regression Analysis,
and Financial Management
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
Outline
2.1 Introduction
 2.2 Financial statement: A brief review
 2.3 Critique of accounting information
 2.4 Static ratio analysis and its extension
 2.5 Cost-volume-profit analysis and its applications
 2.6 Accounting income vs. economic income
 2.7 Summary
 Appendix 2A. Simple regression and multiple
regression
 Appendix 2B. Instrumental variables and two-stage
least squares

2.1
Introduction
Table 2.1
Consolidated
Balance Sheets
of Johnson &
Johnson
Corporation and
Consolidated
Subsidiaries
(dollars in
millions)
Assets
Current Assets
Cash and Cash Equivalent
Marketable Securities
Account Receivable
Inventory
Deferred Taxes on Income
Prepaid Expenses and Other Receivable
Total Current Assets
Marketable Securities – Non0current
Property, Plant and Equipment, net
Intangible Assets, net
Deferred Taxes on Income
Other Assets
Total Assets
Liabilities and Shareholder’s Equity
Current Liabilities
Loans and Notes Payable
Account Payable
Accrued Liabilities
Accrued Salaries, Wages and Commissions
Taxes on Income
Total Current Liabilities
Long-term Debt
Deferred Tax liability
Employee Related Obligations
Other Liabilities
Shareowners’ Equity
Preferred stock-without Par Value
Common Stock – Par Value $1.00
Net Receivable from Employee Stock Plan
Accumulated Other Comprehensive Income
Retained Earnings
Less: Common Stock Held in Treasury
Total Shareowners’ Equity
Total Liabilities and Shareholders’ Equity
2000
2001
2002
2003
$4,278
2,479
4,601
2,905
1,174
1,254
16,691
$3,758
4,214
4,630
2,992
1,192
1,687
18,473
$2,894
4,581
5,399
3,303
1,419
1,670
19,266
5377
4146
6574
3588
1526
1784
22995
657
969
7,409
7,719
7,535
9,077
240
288
1,713
1,962
$34,245 $38,488
121
8,710
9,246
236
2,977
40,556
84
9846
11539
692
3107
48263
46
10436
5979
551
3122
53317
20
10830
6185
385
3221
58025
2004
2005
$9,203 $16,055
3681
83
6831
7010
3744
3959
1737
1845
2124
2442
27320
31394
$1,489
2,122
2,793
529
322
7,255
$565
2,838
3,135
969
537
8,044
$2,117
3,621
3,820
1,181
710
11,449
1139
4966
2639
1452
944
13448
280
5227
3523
1094
1506
13927
668
4315
3529
1166
940
12635
3,120
255
1,804
1,373
2,217
493
1,870
1,631
2,022
643
1,967
1,778
2955
780
2262
1949
2565
403
2631
1978
2017
211
3065
2226
3,120
-35
-461
18,113
342
20,395
3,120
-30
-530
23,066
1,393
24,233
3,120
-25
-842
26,571
6,127
28,824
3120
-18
-590
30503
6146
33015
3120
-11
-515
35223
6004
31813
3120
-755
41471
5965
37871
$34,245 $38,488 440,556
48263
53317
58025
2006
2.2 Financial statement: A Brief Review





Balance Sheet
Income Statement
Retained Earnings Statement
Statement of changes in financial
position
Annual vs. Quarterly Financial Data
Income Statement
Table 2.2: Consolidated Income Statements of Johnson &
Johnson Corporation and Subsidiaries (dollars in millions)
(Dollars in Millions Except Per Share Figures) (Note 1)
Sales to customers
Cost of products sold
Gross profit
Selling, marketing and administrative expenses
Research expense
Purchased in-process research and development (Note 17)
Interest income
Interest expense, net of portion capitalized (Note 3)
Other (income) expense, net
Earnings before provision for taxes on income
Provision for taxes on income (Note 8)
Net earnings
Basic net earnings per share (Notes 1 and 19)
Diluted net earnings per share (Notes 1 and 19)
2000
29,846
8,908
20,938
11,218
3,105
66
-429
204
-94
14,070
6,868
1,915
4,953
1.65
1.61
2001
32,317
9,581
22,736
11,260
3,591
105
-456
153
185
14,838
7,898
2,230
5,668
1.87
1.84
2002
2003
36,298 $41,862
10,447 12,176
25,851 29,686
12,216 14,131
3,957
4,684
189
918
-256
-177
160
207
294
-385
16,560 19,378
9,291 10,308
2,694
3,111
6,597 $7,197
2.2
$2.42
2.16
$2.40
2004
2005
47,348 $50,514
13,422 13,954
33,926 36,560
15,860 16,877
5,203
6,312
18
362
-195
-487
187
54
15
-214
21,088 22,904
12,838 13,656
4,329
3,245
8,509 $10,411
2.87
$3.50
2.84
$3.46
2006
Statement of Equity
Table 2.3:
Consolidated Statements of
Equity of Johnson &
Johnson Corporation and
Subsidiaries (dollars in millions)
Statement of Equity
(cont’d)
Table 2.3:
Consolidated Statements of
Equity of Johnson &
Johnson Corporation and
Subsidiaries (dollars in millions)
(Cont’d)
(Dollars in Millions) (Note 1)
Cash flows from operating activities
Net earnings
Adjustments to reconcile net earnings to cash flows:
Depreciation and amortization of property and intangibles
Purchased in-process research and development
Deferred tax provision
Accounts receivable allowances
Changes in assets and liabilities, net of effects from acquisitions:
Increase in accounts receivable
(Increase)/decrease in inventories
(Decrease)/increase in accounts payable and accrued liabilities
Decrease/(increase) in other current and non-current assets
Increase in other current and non-current liabilities
Statement of Cash Flows
Table 2.4:
Consolidated Statement of
Cash Flow of Johnson &
Johnson Corporation and
Consolidated Subsidiaries,
December 31, 2000,
December 31, 2001,
December 31, 2002,
December 31, 2003,
December 31, 2004,
December 31, 2005,
December 31, 2006.
2001
2002
2003
2004
2005
4,953
5,668
6,597
7,197
8,509 $10,411
1,592
66
-128
41
1,605
105
-106
99
1,662
189
-74
-6
1,869
918
-720
6
2,124
18
-498
3
2,093
362
-46
-31
-468
128
41
124
554
-258
-167
1,401
-270
787
-510
-109
1,420
-1,429
436
-691
39
2,192
-746
531
-111
11
607
-395
863
-568
-396
-911
620
343
6,903
8,864
8,176
10,595
11,131
11,877
Cash flows from investing activities
Additions to property, plant and equipment
Proceeds from the disposal of assets
Acquisitions, net of cash acquired (Note 17)
Purchases of investments
Sales of investments
Other (primarily intangibles)
-1,689
166
-151
-5,676
4,827
-142
-1,731
163
-225
-8,188
5,967
-79
-2,099
156
-478
-6,923
7,353
-206
-2,262
335
-2,812
-7,590
8,062
-259
-2,175
237
-580
-11,617
12,061
-273
-2,632
154
-987
-5,660
9,187
-341
Net cash used by investing activities
-2,665
-4,093
-2,197
-4,526
-2,347
-279
Cash flows from financing activities
Dividends to shareholders
Repurchase of common stock
Proceeds from short-term debt
Retirement of short-term debt
Proceeds from long-term debt
Retirement of long-term debt
Proceeds from the exercise of stock options
-1,724
-973
814
-1,485
591
-35
387
-2,047
-2,570
338
-1,109
14
-391
514
-2,381
-6,538
2,359
-560
22
-245
390
-2,746
-1,183
3,062
-4,134
1,023
-196
311
-3,251
-1,384
514
-1,291
17
-395
642
-3,793
-1,717
1,215
-732
6
-196
696
Net cash used by financing activities
-2,425
-5,251
-6,953
-3,863
-5,148
-4,521
-47
-40
110
277
190
-225
Increase in cash and cash equivalents
Cash and cash equivalents, beginning of year (Note 1)
1,766
2,512
-520
4,278
-864
3,758
2,483
2,894
3,826
5,377
6,852
9,203
Cash and cash equivalents, end of year (Note 1)
4,278
3,758
2,894
5,377
9,203 $16,055
215
1,651
185
2,090
141
2,006
206
3,146
222
3,880
$151
3,429
754
504
971
815
946
131
905
2
802
105
$818
369
241
-5
1,925
-434
550
-72
3,135
-323
595
-15
$1,128
-141
236
-85
151
1,491
-1,266
225
478
–
478
2,812
580
$987
2,812
580
$987
Net cash flows from operating activities
Effect of exchange rate changes on cash and cash equivalents
Supplemental cash flow data
Cash paid during the year for:
Interest
Income taxes
Supplemental schedule of noncash investing and financing activities
Treasury stock issued for employee compensation and stock option plans, net of cash proceeds
Conversion of debt
Acquisitions
Fair value of assets acquired
Fair value of liabilities assumed
Annual vs. Quarterly Financial Data
2000
Net cash paid for acquisitions
Treasury stock issued at fair value
Net cash paid for acquisitions
2006
2.3
Critique of accounting information

Criticism

Methods for improvement
a) Use of Alternative Information
b) Statistical Adjustments
c) Application of Finance and Economic
Theories
2.4
Static ratio analysis and its extension

Static determination of financial ratios
Dynamic analysis of financial ratios
Statistical distribution of financial ratios


Static determination of financial ratios
Table 2.5: Company ratios period 2003-2004
Ratio Classification
Formula
J&J
Industry
2003
2004
2003
2004
Liquidity Ratio
Current Ratio
Current asset
Current liabilities
1.71
1.96
1.59
1.7
Quick Ratio
CA  inventory other CA
Current liabilities
1.21
1.47
1.048
1.174
Debt-to-Asset
Total asset
Total equity
0.44
0.40
0.36
0.35
Debt-to-Equity
Total debt
Total equity
0.80
0.58
1.3
1.45
1.80
1.45
3.61
4.14
12.6
14.6
23.8
27.3
Leverage Ratio
Equity Multiplier
Times Interest Paid
Total debt
Total asset
EBIT
Interest expenses
Static determination of financial ratios
Table 2.5: Company ratios period 2003-2004 (Continued)
Ratio Classification
Formula
J&J
Industry
2003
2004
2003
2004
57.32
52.66
58.3
56.6
6.37
6.93
6.26
6.45
3.39
3.58
3.28
3.42
2.9
2.8
4.5
4.7
0.95
0.92
0.79
0.78
13.2%
15.3%
17.19%
17.97%
14.91%
15.96%
7.34%
7.06%
26.79%
26.75%
14%
12.44%
30.15
24.2
21.35
22.1
5.52
4.68
5.71
5.92
Activity Ratios
Average collection period
Accounts receivable Turnover
Inventory Turnover
Fixed Asset Turnover
Total Asset Turnover
Profitability Ratios
Profit margin
Return on assets
Return on equity
Account Re ceivable
Sales / 365
Sales
AcountsRe ceivable
Cost of Good Sold
Inventory
Sales
Fixed assets
Sales
Total assets
Net income
Sales
Net incom e
Total assets
Net Incom e
Total equity
Market value
Price/earnings
Price-to-book-value
Market price per share
Earning per share
Market price per share
Book value per share
Dynamic Analysis of Financial Ratios
Yj ,t  Yj ,t 1   j (Yj*,t  Yj ,t 1 ),
(2.1)
where
0j1, and
j = A partial adjustment coefficient;
Yj,t = Firm’s jth financial ratio period t;
Yj,t-1 = Firm’s jth financial ratio period t-1; and
Y*j,t = Firm’s jth financial ratio target in period t,
Dynamic Analysis of Financial Ratios
Yj*,t  CX j ,t 1   j ,t
(2.2)
Yj ,t  Yj ,t 1   j [ X j ,t 1  Yj ,t 1 ]
(2.3)
Z j ,t  Aj  BjWj ,t 1   j ,t
(2.4)
where
Zj,t = Yj,t - Yj,t-1;
Wj,t-1 = Xj,t-1 - Yj,t-1;
Aj and Bj = Regression parameters,
and
j,t = The error term.
Dynamic Analysis of Financial Ratios
Z′j,t = A′j + B′jW′j,t-1 + ′j,t,
where
Z′j,t = log (Yj,t) - log (Yj,t-1);
W′j,t-1 = log (Xj,t-1) - log (Yj,t-1);
and
′j,t = The Error term.
(2.5)
Dynamic Analysis of Financial Ratios
Bj 

 log(Y j ,t / Y j ,t 1 )
 log( X j ,t 1 / Y j ,t 1 )
% change in [Yj,t / Y j ,t 1 ]
% change in [X j,t-1 / Y j ,t 1 ]
Yj,t*  CX j ,t 1
(2.6)
(2.7)
Yj ,t  Aˆ  Bˆ1 X j ,t 1  Bˆ2Yj ,t 1   j ,t
(2.8)
Dynamic Analysis of Financial Ratios
Table 2.6: Dynamic adjustment ratio regression results
Variable
Current Ratio
Leverage Ratio
Mean Z
0.0075
-0.03083
Mean W
-0.14583
0.361666667
Var(Z)
0.013039
0.006099
Cov(Z,W)
0.074
0.009
Bj`
0.810*
0.259
t-Statistics
[3.53]
[1.06]
Aj`
0.032
-0.042
* Partial adjustment coefficient significant at 95% level
Dynamic Analysis of Financial Ratios
Table 2.7: Ratio correlation coefficient matrix
CR
CR
AT
GPM
LR
AT
GPM
LR
1.0
-0.443841
1.0
0.363273
0.381393
1.0
-0.51175
0.21961
-0.05028
1.0
Dynamic Analysis of Financial Ratios
Z1,t = A0 +A1Z2,t + A2W1 + 1,t, (2.9a)
Z2,t = B0 + B1Z1,t + B2W2 + 2,t. (2.9b)
where
Ai, Bi (i = 0, 1, 2) are coefficients, 1 and 2 are error terms,
and
Z1,t = Individual firm’s current ratio in period t
- individual firm’s current ratio in period t-1;
Z2,t = Individual firm’s leverage ratio in period t
- individual firm’s leverage ratio period t-1;
W1,t = Industry average current ratio in period t-1
- individual firm’s current ratio period t-1;
W2,t = Industry average leverage ratio in period t-1
- individual firm’s leverage ratio in period t-1.
Dynamic Analysis of Financial Ratios
Table 2.8: Johnson & Johnson empirical results for the simultaneous
equation system
A0(B0)
A1(B1)
A2(B2)
(2.9a)
-0.071
[-1.80]
-0.378
[-5.52]
0.080
[1.20]
(2.9b)
-0.0577
[-1.59]
-0.842
[-6.07]
0.074
[0.91]
Statistical Distribution of Financial Ratios
1
 ( X   )2 / 2 2
F[ X ] 
e
 2
(    X  ),
(2.10)
where  and 2 are the population mean and variance, respectively, and
e and  are given constants; that is,  = 3.14159 and e = 2.71828.
Statistical Distribution of Financial Ratios
There is a direct relationship between the normal
distribution and the log-normal distribution. If Y is lognormally distributed, then X = log Y is normally distributed.
Following this definition, the mean and the variance of Y
can be defined as:
1 2
Y  exp(x   x ),
2
Y2  exp(2x   x2 )(exp( x2 )  1),
(2.11a)
(2.11b)
where exp represents an exponential with base e.
Statistical
Distribution of
Financial Ratios
2.5 COST-VOLUME-PROFIT ANALYSIS AND ITS
APPLICATIONS


Deterministic analysis
Stochastic analysis
2.5.1 Deterministic Analysis
Operating Profit = EBIT = Q(P－V)－F,
where
Q = Quantity of goods sold;
P = Price per unit sold;
V = Variable cost per unit sold;
F = Total amount of fixed costs; and
P - V = Contribution margin.
(2.12)
2.5.1 Deterministic Analysis (cont’d)
If operating profit is equal to zero, Eq. (2.12) implies that Q(P-V)-F=0 or that
Q(P-V)=F, that is,
Q* 
F
(P  V )
(2.13)
Equation (2.13) represents the break-even quantity, or that quantity of sales at
which fixed costs are just covered.
The definition of the degree of operating leverage (DOL) is,
% Change in profits
Q( P  V )
Fixed Costs
DOL 

=1 
% Change in sales Q( P  V )  F
Profits
(2.14)
Based upon the definition of linear break-even quantity defined in Eq. (2.13),
the degree of operating leverage can be rewritten as
1
DOL 
.
*
[1  (Q / Q)]
(2.15)
2.5.2 Stochastic Analysis
In reality, net profit is a random variable because the quantity used in the
analysis should be the quantity sold, which is unknown and random, rather than
the quantity produced, which is internally determined. This is the simplest form
of stochastic CVP analysis; for there is only one stochastic variable and one
need not be concerned about independence among the variables. The
distribution of sales is shown graphically in Fig. 2.5.
2.6 ACCOUNTING INCOME VS. ECONOMIC
INCOME
Et = At + Pt,
where
Et = Economic income,
At = Accounting earnings,
and
Pt = Proxy errors.
(2.17)
2.7 SUMMARY
In this chapter, the usefulness of accounting information in
financial analysis is conceptually and analytically evaluated.
Both statistical methods and regression analysis techniques are
used to show how accounting information can be used to
perform active financial analysis for the pharmaceutical industry.
In these analyses, static ratio analysis is generalized to dynamic
ratio analysis. The necessity of using simultaneous-equation
technique in conducting dynamic financial ratio analysis is also
demonstrated in detail. In addition, both deterministic and
stochastic CVP analyses are examined. The potential
applications of CVP analysis in financial analysis and planning
are discussed in some detail. Overall, this chapter gives readers
a good understanding of basic accounting information and
econometric methods, which are needed for financial analysis
and planning.
Appendix 2A. Simple regression and multiple regression
2. A.1 INTRODUCTION
2. A.2 SIMPLE REGRESSION
Variance of bˆ
Multiple Regression
Appendix 2A. Simple regression and multiple regression
Yt  a  bX t 1   t
(2.A.1a)
log Yt  a  b log X t 1   t
(2.A.1b)
Var Yt   Var a  bX t 1  t 
 Var a  Var bX t 1   Var  t   2cov a, bX t 1   2cov a,  t   2cov bX t 1, t ,
(2.A.2a)
Var Yt   b2Var  X t 1   Var t 
(2.A.2b)
Appendix 2A. Simple regression and multiple regression
Variation explained by the explanatory variable
R 
Total variation in the dependent variable
2

b 2Var  X t 1 
(2.A.3)
Var Yt 
2
n
n
ˆ

ESS   Yt  Yˆt    Yt  aˆ  bX
t 1

t 1
t 1 

2

n
 ( ESS )
ˆ
 2  Yt  aˆ  bX
t 1  0
t 1
a

(2.A.4)
(2.A.5a)

n
 ( ESS )
ˆ
 2  X t 1 Yt  aˆ  bX
t 1  0
t 1
b
(2.A.5b)
Appendix 2A. Simple regression and multiple regression
n
n
t 1
t 1
ˆ  bˆ  X t 1   Yt
an
n
n
(2.A.6a)
n
aˆ  X t 1  bˆ  X t21   X t 1Yt
t 1
t 1
t 1
(2.A.6b)
Appendix 2A. Simple regression and multiple regression
n
 Yt
n
n
bˆ 
 X t 1
t 1
n
n
 X t 1
t 1
t 1
n
 X t 1Yt
t 1
n
 X t 1
t 1

n
n
n
t 1
t 1
n
t 1
n
t 1
t 1
n(  X t 1Yt )  (  X t 1  Yt )
(2.A.7)
n  X t21  (  X t 1 ) 2
n
 X t21
t 1
Cov[ X t 1 , Yt ]
ˆ
b
Var[ X t 1 ]
(2.A.7a)
Appendix 2A. Simple regression and multiple regression
Y
 X t 1
n
n
n
n
2
(
Y
)(
X
)

(
X
)(
 t  t 1
 t 1  X t 1Yt )
 X t 1Yt  X t21
t 1
t 1
t 1
aˆ 
 t 1
n
n
n
 X t 1
n  X 2  ( X )2
 X t 1

X
t 1
2
t 1
n
n
n
t 1
t 1
t 1
t 1
t 1
t 1
n
n
n
n
t 1
t 1
t 1
(  Yt n)[n(  X t21 )  (  X t 1 ) 2 ]  (  X t 1 n)[n(  X t 1Yt )  (  X t 1 )(  X t 1 )]
t 1
n
n
t 1
t 1
(2.A.8)
n  X t21  (  X t 1 ) 2
aˆ  Y  Xbˆ
(2.A.8a)
Variance of bˆ
Equation (2.A.7a) implies that:
n
(
x
y
)
t

1
t
bˆ   n 2   Wt 1 yt
t 1  t 1 xt 1
t 1
n
Where
xt 1  X t 1  X
yt  Yt  Y
xt 1
Wt 1  n 2
 t 1 xt 1
(2.A.7b)
bˆ
Variance of
n
n
t 1
t 1
bˆ   Wt 1bxt 1   Wt 1 t
(2.A.7c)
Var (bˆ)  E (bˆ  b) 2
n
n
t 1
t 1
 E (  Wt 1bX t 1   Wt 1 t  b)2
n
n
t 1
t 1
 E[(  Wt 1 xt 1  1)b   Wt 1 t ]2
n
 E (  Wt 1 t ) ,
2
t 1
n
since  Wt 1 xt 1  1.
t 1
Var(bˆ)  E[(W01 )2  2(W0W11 2 )  (W1 2 )2  ] (2.A.9)
Variance of
bˆ
Var (bˆ)  E (W01 )2  E (W1 2 )2 
 W02 E (12 )  W12 E ( 22 ) 
n
Var (bˆ)   Wt 21 E ( t2 )
t 1
n
  2  Wt 21
t 1
n
W
t 1
2
t 1
1
 x

 n 2
( x )
t 1 xt 1
n
2
t 1 t 1
n
2
2
t 1 t 1
Variance of bˆ
Var (bˆ) 
 2
 x
n
2
t 1 t 1
 x
Var (aˆ )   
n x
2
Cov(aˆ, bˆ)   2
(2.A.10)
n
2
t 1 t 1
n
2
t 1 t 1
(2.A.11)
X
tn1 xt21
(2.A.12)
Multiple Regression
Yt  a  bX1,t 1  cX 2,t 1  t
(2.A.13a)
The error sum of squares can be defined as:
2
ˆ
ESS   ˆ  (Yt  Yt )
2
t
Where
ˆ
ˆ
Yˆt  aˆt  bX
1,t 1  cX 2,t 1
Multiple Regression
ESS
 0 or  Yt  na  b  X 1,t 1  c  X 2,t 1
a
(2.A.14a)
ESS
 0 or  X 1,t 1Yt  a  X 1,t 1  b  X 1,2t 1  c  X 1,t 1 X 2,t 1
b
(2.A.14b)
ESS
 0 or  X 2,t 1Yt  a  X 2,t 1  b  X 1,t 1 X 2,t 1  c  X 2,2 t 1
c
(2.A.14c)
Multiple Regression
0 = na + b(0) + c(0),
(2.A.15a)
 x1,t 1 yt  a(0)  b  x1,2t 1  c  x1,t 1x2,t 1 (2.A.15b)
 x2,t 1xt  a(0)  b  x1,t 1x2,t 1  c  x
2
2,t 1
(2.A.15c)
Multiple Regression
bˆ 
2
x
y
(
x
 1,t 1 t  2,t 1 )   x2,t 1 yt  x1,t 1 x2,t 1
( x12,t 1 )( x 22,t 1 )  ( x1,t 1 x 2,t 1 ) 2
(2.A.16a)
2
 x2,t 1 yt ( x1,t 1 )   x1,t 1 yt  x1,t 1 x2,t 1
cˆ 
(2.A.16b)
2
2
2
( x1,t 1 )( x2,t 1 )  ( x1,t 1 x2,t 1 )
ˆ  cX
ˆ 2
aˆ  Yˆ  bX
1
(2.A.17)
Multiple Regression
Yt  0.2837  0.7564 X 1,t 1  0.2990 X 2,t 1
(0.4323)
(0.3288)
(0.2240)
(2.A.13b)
aˆ  1.7071(0.7564)(1.8448)(0.2990)(1.6904)
 0.2837
(Yt  Yt )  (Yt  Yˆt )  (Yˆt  Yt )
ˆ
ˆ
Yˆt  aˆ  bX

cX
1,t 1
2,t 1
(2.A.18)
(2.A.19)
Multiple Regression
(Yt  Yt ) 2  (Yt  Yˆt ) 2  (Yˆt  Yt ) 2 , (2.A.20)
TSS
ESS
RSS
where
TSS = Total sum of squares;
ESS = Residual sum of squares; and
RSS = Regression sum of squares.
Multiple Regression
2
2
ˆ
ˆ

(
Y

Y
)


RSS
t
t
t
R2 


1

TSS (Yt  Yt )2
(Yt  Yt )2
2
ˆ


t
R2  1 
Var (Yt )
(2.A.21)
(2.A.22)
where
Var (ˆt )    2
 ˆt 2

nk
(Yt  Y )2
Var (Yt ) 
n 1
and k = the number of independent variables.
Multiple Regression
n 1
R  1  (1  R )
nk
2
2
(2.A.23)
R nk
F (k  1, n  k ) 
1  R2 k 1
2
where F(k-1, n-k) represents F-statistic with
k－1 and n－k degrees of freedom.
Appendix 2B. Instrumental Variables and TwoStage Least Squares
2. B.1 ERRORS-IN-VARIABLE PROBLEM
2. B.2 INSTRUMENTAL VARIABLES
2. B.3 TWO-STAGE, LEAST-SQUARE
2. B.1 ERRORS-IN-VARIABLE PROBLEM
R j ,t  A j  B j Rm,t   t
(2.B.1)
R*m,t  Rm,t  Vt
(2.B.2)
Var(R )  Var(Rm,t  Vt )    
*
m,t
2
m
2
V
(2.B.3)
2. B.1 ERRORS-IN-VARIABLE PROBLEM
R j ,t  Aj  B j R
Bˆ j 

Cov( Rm* ,t , R jt )
*
m ,t
Var ( R )

*
m,t

*
t
(2.B.4)
Cov( Rm,t  Vt ,  j  B j Rm ,t   t )
Var ( Rm,t )  Var (Vt )
B j Cov( Rm,t , Rm ,t )  Cov(Vt ,  t )
Var ( Rm,t )  Var (Vt )

Bj
1   V2 /  M2
(2.B.5)
2. B.2 INSTRUMENTAL VARIABLES
Cov(Rj , Z )  Bj Cov(Rm , Z )  Cov(Z ,  )
Bˆ j 
Cov( R j , Z )
*
m
Cov( R , Z )

Cov( R j , Z )
Cov( Rm , Z )
Y1  A0  A1Y2  E1
Y2  B0  B1Y1  B2 Z1  E2
(2.B.6)
(2.B.7)
(2.B.8a)
(2.B.8b)
2. B.2 INSTRUMENTAL VARIABLES
Y1  A0  A1Y2  A2 Z 2  E1
Y2  B0  B1Y1  B2 Z1  E2
(2.B.9a)
(2.B.9b)
Y1  A0  AY
1 2  A2 Z2  A3 Z3  E1Y (2.B.10a)
Y2  B0  B1Y1  B2 Z1  E2
(2.B.10b)
2.B.3 TWO-STAGE LEAST-SQUARE
Y1  C0  C1 Z1  C2 Z 2  C3 Z 3  E1
(2.B.11a)
Y2  D0  D1 Z1  D2 Z 2  D3 Z 3  E2
(2.B.11b)
Y1  A0  A1Yˆ2  A2 Z 2  A3 Z 3  E1
(2.B.10′a)
Y2  B0  B1Yˆ1  B2 Z1  E2
(2.B.10′b)
2.B.3 TWO-STAGE LEAST-SQUARE
Y1  0.2399  0.8198Z1 1.9004Z1 , R2  0.3449,
(0.1012) (0.2802) (1.245)
(2.B.12a)
Y2  0.0746  0.1133Z1  0.7849Z2 , R2  0.4240,
(0.0195) (0.0541) (0.2405)
(2.B.12b)