Gurumurthy Kalyanaram

Prior work in marketing has suggested that advertising threshold effects—levels beneath which there is essen-tially no sales response—are rarely encountered in practice. Because advertising policies settle into effective ranges through early trial and error, thresholds cannot be observed directly, and arguments for their existence must be based primarily on a “statistical footprint,” that is, on relative fits of a range of model types. To detect possible threshold effects, we formulate a switching regression model with two “regimes,” in only one of which advertising is effective. Mediating the switch between the two regimes is a logistic function of category-specific dynamic variables (e.g., order of entry, time in market, number of competitors) and advertising levels, nesting a variety of alternative formulations, among them both standard concave and S-shaped responses. A sequence of comparisons among parametrically related models strongly suggests: that threshold effects exist; that market share response to advertising is not necessarily globally concave; that superior fit cannot be attributed to model flexibility alone; and that dynamic, environmental, competitive, and brand-specific factors can influence adver-tising effectiveness. These effects are evident in two evolving durables categories (SUVs and minivans), although

not in the one mature, nondurable category (liquid detergent) studied.

Key words: advertising; econometric models; product management; switching regression; dynamic models History: This paper was received April 26, 2000, and was with the authors 17 months for 3 revisions; processed by William Boulding.

Introduction

Advertising threshold effects, those observed when “some positive amount of advertising is necessary before any sales impact can be detected” (Hanssens et al. 2001, p. 113), have received a good deal of attention in the marketing literature (Lambin 1976, Bemmaor 1984). Although managers and advertising practitioners often profess some degree of belief in advertising threshold effects (Corkindale and Newall 1978, Ambler 1996), empirical evidence for their existence has been limited (Hanssens et al. 2001, Vakratsas and Ambler 1999). This raises the question of whether advertising thresholds exist and, if so, why they have proved difficult to observe or measure.

Several lines of argument have been developed to account for this paucity of evidence. The first, stemming from Bemmaor (1984), is that much of theevidence “rejecting” the existence of a threshold effect is based on the adequate fit of a globally con-cave advertising response function (e.g., a double-log advertising response function, as in Lambin 1976), which implies decreasing returns to advertising, as opposed to the overt rejection of, say, an S-shaped response function. A second argument (Bronnenberg 1998, Simon and Arndt 1980, Steiner 1987, Vakratsas and Ambler 1999) proposes that most studies of advertising response have concerned themselves with frequently purchased, mature product categories, where the competitive environment is stable, advertis-ing budgets are set, and the operating range of adver-tising expenditures for brand managers lies above the threshold point.

By contrast, given the situation typifying new-brand introductions, especially in emerging product

Vakratsas, Feinberg, Bass, and Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS

categories—a dynamic competitive environment, threats of competitive entry, active awareness build-ing, high consumer uncertainty, and low expertise— thresholds may stand, relatively speaking, in bold relief. To this end, we develop a dynamic model of advertising response that encodes thresholds in a (latent) probabilistic manner in the style of Bemmaor (1984), but includes the type of dynamic decision and environmental variables considered in numerous prior studies (cf. Bowman and Gatignon 1996). In con-trast to those studies, we apply the model to data on two evolving, durable categories as well as to one fre-quently purchased, nondurable one.

The model to be developed allows formal tests for the existence of threshold effects, for whether they are dynamic (time-varying), and for whether they vary across brands, in precisely the type of situa-tion in which they are most likely to be detected. While several of these goals have been addressed by prior research, there has often been a presumption that threshold effects are either static, nonexistent, or unimportant, vantage points that the present study seeks to redress.

Model Formulation and Discussion of Conjectures

As suggested by Bemmaor (1984), advertising thres-hold effects, as well as a method for their estima-tion, may be detected by recourse to a mixture of two concave advertising response functions; that is, a two-regime market share model. In one regime, where one operates below the critical threshold, advertis-ing effects are negligible, whereas in the second regime advertising is (measurably) effective so that expenditures and share can be related through any of a number of monotonic functional forms. We call these, respectively, the “ineffective” and “effective” regimes. In real markets, demarcation points between the two regimes are difficult to pin down, let alone directly observe. Lack of direct threshold observabil-ity can be accommodated through a (latent) prob-abilistic mechanism, while the market share model in each of the two regimes takes a multiplicative form; prior research has confirmed the multiplicative specification to perform essentially as well as attrac-tion models in accounting for market share effects (Brodie and de Kluyver 1984, Ghosh et al. 1984). We have deliberately called upon model formulations that have been widely validated empirically (such as the multiplicative and that of Bemmaor), and any claim to methodological novelty lies in their combi-nation and joint estimation in a dynamic context.

If advertising thresholds indeed exist, the likeli-hood or “probability” of shift between the ineffective and effective regimes should clearly depend on adver-tising expenditure levels. We highlight the term “probability” because it is important to note that the two-regime model operates in a manner similar to, for example, mixture models: While the model is itself formally probabilistic, the nature of response is not assumed to be.

Specifically, the model takes the following form:

Regime 1, “ineffective:” M1it = eK1 Dit_D Pit_P Ei_E Tit_T eu1it

with probability pit (1)

Regime 2, “effective:” M2it = eK2 A_itA Dit_D Pit_P Ei_E Tit_T eu2it with probability 1 − pit _

where, for brand i at time t:

Mit = Market share

Kr = Intercept (constant) term for regime r

Ait = Advertising expenditures

Dit = Distribution (i.e., number of outlets)

Pit = Price

Ei = Order of entry

Tit = Time in market 2

urit = Error term for regime r, N _0_ _r _.

Note that the specification for the two regimes is identical up to estimated coefficients once that for Advertising _Ait _ is set to zero for the “ineffective” regime, as it must be by definition.

Although the share model for each regime implies diminishing returns to advertising spending, the probabilistic nature of the switching mechanism sug-gests a region where returns to advertising are increasing, with location depending on pit .2 The mar-ket share formulation (1) does not address p , the prob-ability of falling into the ineffective regime, which can take a variety of forms. For example, p might lack a direct functional relationship to any quantities of managerial interest, and so would be zero (so that there is no ineffective regime) or a nonzero constant (either estimated or prespecified). Prior models have invoked such restrictions on p, and we will test the proposed model against them.

Aside from advertising itself, several specific factors—number of competitors, order of entry, and

1 “Time in Market” for any particular brand is perfectly collinear with time, and so captures temporal trends on a brand-by-brand basis. Brand-specific constants have not been estimated, as there is an overall regime-specific constant and both brand-specific order-of-entry and time-in-market values. Bowman and Gatignon (1996), who used an equivalent formulation, discuss the rationale for doing so (p. 236).

2 We refer to pit as p where no ambiguity arises. Although this may seem equivalent to an S-shaped response function, it is consider-ably more flexible. It is well established (Little 1979, Mahajan and Muller 1986, Feichtinger et al. 1994, Feinberg 2001) that advertising threshold effects influence the shape of the advertising response function, and thereby affect the qualitative nature of the optimal advertising policy.

Vakratsas, Feinberg, Bass, and Gurumurthy Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS time in market—can enhance or diminish advertis-ing expenditure effects. In the present framework they can be taken to affect advertising thresholds via the regime-switching probability, p. For the purposes of estimation, pit is expressed as a logistic function of the relevant input variables:

pit =_1+exp_−_c0 +c__it +c_ _it +cN Nit +cE Ei___−1 (2)

where, at time t and for brand i, _it = log_Ait _, Nit = number of competitors, and _it = log_Tit _.3 The system (1)–(2) incorporates dynamic elasticities in much the same manner as that suggested by Bowman and Gatignon (1996), only in our case advertising elasticities are affected by control variables (order of entry, etc.) in a potentially nonmonotonic manner.

In sum, the proposed model allows one to model and test for the existence of dynamic, brand-specific advertising thresholds, as well as to obtain advertising elasticities which account for brand, cate-gory, and competitive dynamics.

Comparison with Extant Model Formulations

The proposed probabilistic thresholds model of (1)–(2) reduces to Bemmaor’s (1984) model, when pit does not admit of explanatory variables. A relevant restricted subcase has p = 0, a single regime with globally concave response to all input variables. This “classical” specification is useful as a benchmark, to help gauge the additional explanatory power of a second, ineffective regime, irrespective of whether p admits of decision variables (as in the proposed model) or not (as in Bemmaor’s). Another important special case is one for which the regime-switching probability p is not constant, but follows a distribu-tion with stable, estimated parameters, as in the one-dimensional cutoff (grid search) model:

1 if A A_

p it > _ (3)

it =

A 0 otherwise_

where _ is the advertising threshold. The cutoff model presumes that although there is indeed a switch between the two regimes, it is static, determin-istic, homogeneous (across brands), single-attribute/ unidimensional, and that one operates with perfect certainty in either the effective or ineffective regime.

3 Here we use the log form for advertising expenditure and time in market due to the wide range of these variables, to avoid claiming an overly good fit due to outlier leverage. Alternate runs based on using the nonlogged forms yielded appreciably inferior results in this regard.

4 Specifically, the model given by (1) and (2) implies share elasticity to advertising is (proof available from authors) _it = _1 − p___Mit −

M2it _c_ + M2it _A_/Mit , where Mrit is the share for brand i at time t under regime r and Mit = pM1it + _1 − p_M2it .

Empirical Issues

In the proposed model, advertising threshold effects correspond to parameters _c__ c_ _ cN _ cE _ which medi-ate advertising’s main effect on switching probability and its dependence on, respectively, ad expenditure level, time in market (dynamic factor), number of competitors (dynamic factor), and order of entry (brand-specific factor). We briefly consider these “main effects” in turn. A considerably more detailed account is provided by Bowman and Gatignon (1996) and Parker and Gatignon (1996).

Advertising Expenditure Level. Existence of a threshold suggests that, ceteris paribus, a higher level of advertising serves to reduce the probability _p_ of being in the ineffective regime. One therefore expects c_ to be negative.

Time in Market. If a brand tends to advertise at all, the longer it has been in the market, the greater the likelihood that consumers have been repeatedly exposed to its messages. For established brands, the role of advertising becomes largely one of reminding consumers: reinforcing knowledge of brand attributes, rather than “informing” (Batra et al. 1996, Smith and Swinyard 1982, Smith 1993). In such cases, lower advertising levels may be sufficient to surpass the threshold, that is, to be within the effective regime. Thus, the longer a brand has been in the market, the lower its advertising thresholds are likely to be, and so c_ is expected to be negative.

Number of Competitors. Where noise is greater, so too is the effort one must expend to be heard above it (Webb 1979, Webb and Ray 1979). Specifically, a larger number of direct competitors should translate into elevated advertising thresholds, resulting in higher below-threshold probabilities, so that cN is expected to be positive.

Order of Entry. Earlier entrants often serve as cat-egory prototypes (Carpenter and Nakamoto 1989) and tend to benefit more from advertising outlays, as information learned about them is treated as novel by consumers (Kardes and Kalyanaram 1992). Advertising of early entrants can therefore serve as a barrier to entry for later competitors (Bain 1956, Comanor and Wilson 1967) in the form of higher advertising thresholds, so that cE is expected to be positive.

The four posited directional results _c_ < 0, c_ < 0, cN > 0, cE > 0_ can combine to render some com-parisons indeterminate. For example, while c_ < 0 suggests that the longer a brand has been in the market the lower advertising thresholds should be, cE > 0 suggests that an increase in the number of competitors should lead to an increase in advertis-ing thresholds. Over time, a brand’s in-market dura-tion and its number of competitors both tend to

Vakratsas, Feinberg, Bass, and Gurumurthy Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS increase, yet their directional effects on thresholds are predicted to oppose one another. Put more directly, although the effects of each explanatory variable (on the below-threshold probability, p) are assumed monotonic in the logistic formulation (2), their inter-actions may well lead to nonmonotonic relationships.

Empirical Application

Data

As discussed at the outset, confirming advertising threshold effects may not be possible in categories for which data are most readily available: frequently pur-chased, low-cost goods from mature categories with stable ad budgets. Thus, for empirical purposes, we have made use of data from two durable product categories: For “Category 1” (Sport Utility Vehicles, 26 brands over 10 years), data begin five years after its introduction, with nine competitors already in the market; in “Category 2” (passenger minivans; 16 brands over 10 years), data begin at category incep-tion. For comparison, we also estimate the model for a stable, frequently purchased category, liquid laun-dry detergent (“Category 3”). Over the observation period (1998–2000), there were 15 major brands, with no entries or exits. In our discussions, we will focus primarily on the durables categories, as numerous prior studies have examined frequently purchased consumer goods; however, we examine model impli-cations for all three categories.

Despite outward differences, both durables cate-gories (during the period covered by the data) are highly dynamic: They undergo considerable changes in terms of the overall number of competitors (and consequently their relative market-timing measures), as well as levels of their various marketing-mix activ-ities. The data for all three categories consist of infor-mation on market share, advertising expenditures, price, distribution (number of outlets), time in market, order of entry, and the number of competitors at any given time, measured here in monthly intervals.5 Finally, advertising information was provided quar. Top Five advertisers than Category 1 (SUVs). This suggests that there are fewer proverbial big spenders in Category 2, yet who each appear to spend more, and with greater consistency (lower coefficient of vari-ation), than their Category 1 counterparts. Further, there are fewer competitors overall in Category 2, due perhaps to its being at an earlier stage of devel-opment. Regardless, we would anticipate that either of the durables categories (1 or 2) would provide a clearer window through which to observe threshold effects than nondurable Category 3.

Estimation and Empirical Results

The joint system (1)–(2) was estimated through maxi-mum likelihood implemented in Gauss.6

Due to substantial differences between the esti-mated parameters for the two categories, the empir-ical analysis was performed separately for each. Parameter estimates for the model given by (1)–(2) appear in Table 1. We discuss the four posited effects in turn.

Advertising Threshold Effects. Advertising effects on the regime-switching probability are, as conjec-tured, negative and significant for all three cate gories _c_1 = −0_45; c_2 = −0_95; c_3 = −0_19_.7 A higher ad-expenditure level therefore increases the probability of surpassing the threshold, and thus operating in the effective regime. We stress that this outcome is in no way guaranteed by the form of the model or the estimation method. In fact, a closer look at Category 3 (liquid detergent) indicates that although advertising level affects switching probabil-ity p, there are nevertheless no dvertising-based threshold effects, for the following reason. In the durables cate-gories, advertising coefficients in the effective regime are strongly distinguishable from zero __A1 = 0_32, t = 24_4; _A2 = 0_24, t = 10_6_. By contrast, in the non-durable category, advertising has negligible effect in 6 The market share density for brand i at time t (for the standard normal density, pit given by (3)) and associated log likelihood, are: terly in terms of dollar expenditures (in units of thousands) and covers print media and television advertising; within each quarter, a uniform distribu-tion is used to convert to monthly values.

The durables categories differ in terms of advertis-ing deployment: Category 2 (minivans) has a higher average expenditure level and a far greater propor-tion of advertising spending concentrated among the

fit = f _log_Mit _ _ __ __ c_ __

_ log_Mit _ − x_ _ _ + _1 − pit _

= pit 1it

_

1

where:

x1_it = _K1 ln Dit ln Pit ln Ei ln Tit __

x2_it = _K2 ln Ait ln Dit ln Pit ln Ei

__ = __D _P _E _T __ __ = __A

_ log_Mit _ − x2_it _ __ _2

ln Tit __

_D _P _E _T __

5 “Price” denotes vehicle base price as published annually by Ward’s Automotive, irrespective of dealer incentives, rebates, or optional equipment; see Bowman and Gatignon (1996, p. 230) for more detail. Quarterly data on “quality” were also available but because they did not prove significant for either category in the forthcoming exploratory model analysis, we make no further reference to it.

T I

c_ = _c0 c_ c_ cN cE __log # = _t=1 _i=1 log_fit __

7 Numerical subscripts, other than for constants K1, K2, and c0, denote category number. Significance levels for all reported param-eters appear in Table 1.

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Table 1 Switching Regime Model Parameter Estimates (t-Statistics)

Category 1: Sport Utility Category 2: Passenger Category 3: Liquid Laundry

Vehicles Minivans Detergent

Regime 1 Regime 2 Regime 1 Regime 2 Regime 1 Regime 2

Variable Ineffect. Effective _ Ineffect. Effective _ Ineffect. Effective _

Intercept∗ 4.33 — 8.59 — 7_35 1_28 6_07

(5.8) (5.2) _17_6_ _0_52_ _3_4_

Advertising — 0_32 0_32 — 0_24 0_24 — 0_005 0_005

__A A _ _24_4_ (24_4) _10_6_ (10_6) −1_5 _0_5_ (0_5)

Distribution 0_96 0_15 0_81 1_12 0_61 0_51 0_43 −1_93

__D D _ _26_0_ _4_1_ _22_0_ _8_5_ _12_0_ _5_1_ _−12_2_ _2_4_ _−12_6_

Price −1_57 −0_85 −0_72 −2_13 −1_82 −0_31 0_21 0_46 −0_25

__P P _ _−13_3_ _−22_5_ _−8_2_ _−9_5_ _−10_5_ _−1_5_ _9_7_ _5_1_ _−3_8_

Order of entry 0_11 0_15 −0_04 −0_32 −0_39 0_07 −0_19 −0_35 0_16

__E E _ _1_6_ _0_8_ _−0_5_ _−2_0_ _−6_2_ _0_6_ _−4_7_ _−0_7_ _0_5_

Time in market −0_13 0_42 −0_30 −0_05 0_21 −0_26 −0_07 −0_55 0_48

__T T _ 2 _−2_6_ _12_7_ _−13_0_ _−0_4_ _4_1_ _−2_7_ _−1_1_ _−1_5_ _1_8_

Variance, _ 1_10 0_48 1_89 0_52 0_14 0_44

Probability of Regime Switch∗∗

Intercept −2_52 9_49 −0_71

_−2_64_ _5_1_ _−2_4_

Advertising, c_ –0_45 –0_95 –0_19

(–10_0) (–46_2) (–5_9)

Time in −0_29 −2_51 —

market, c_ _1_4_ _−4_8_

Number of 0_14 0_52 —

competitors, cN _4_6_ _3_9_

Order of entry, cE 0_07 −0_64 —

_1_6_ _−4_3_

Number of 1 973 960 420

observations −2 682_3 −1 245_5 −122_59

Log likelihood

Notes. ∗Because the intercepts in the two regimes did not differ significantly for either Category 1 or 2, they were estimated as a single value for each. ∗∗For Category 3, three effects _c_ cN cE _ did not approach significance, and so were set to zero for re-estimation.

The second regime __A3 = 0_005, t = 0_5_. Appar-ently, advertising level simply mediates, through p, regime differences in the other dynamic variables, distribution, price, and order of entry. This is espe-cially intriguing, as only advertising has nonnegligi-ble influence in the effective regime itself (i.e., c_ , cN , and cE are nonsignificant).

8 In terms of the dynamic and brand-specific effects on the below-threshold probability, the durable and nondurable categories differ markedly. Time in market is in the conjectured direction for both dura-bles categories, though significantly so only for Category 2, the “younger” of the two _c_1 = −0_29; c_2 = −2_51; c_3 ≡ 0_. The proper interpretation of this for Category 2 is that the longer a brand is in the mar-ket, the higher its probability of being above thresh-old (for a given level of advertising expenditures), so that “older” brands have lower thresholds, all else being equal. As conjectured, number of competitors is significantly positive for both durables categories _cN 1 = 0_14; cN 2 = 0_52; cN 3 ≡ 0_, suggesting that a larger number of competitors decreases the probabil-ity of surpassing the threshold.

Contrary to conjecture, order of entry effects either fail to significantly affect regime-switching probabi-lity _cE1 = 0_07, cE3 ≡ 0_ or do so in a negative manner _cE2 = −0_64_. In other words, later entrants appear to have higher probability of being above threshold, at least in Category 2. A plausible explanation concerns the possibility of later entrants’ capitalizing on the “awareness-building” expenditures of those before them.9 Still, we consider this an anomalous, if equiv-ocal, finding.

Considering these effects for time in market and number of competitors, the data suggest that, at least for Category 2 (in which both significantly affect below-threshold probability), there is a trade-off of sorts taking place. Whereas the “time in market effect” suggests that thresholds decrease the longer a brand is in the market, the “number of competi-tors effect” serves to increase thresholds. Therefore,

As such, we present model results for these three _c_ , cN , and cE _ constrained to zero in Category 3.

We are grateful to a reviewer for pointing this out.

Vakratsas, Feinberg, Bass, and Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS as categories evolve—and both time in market and number of competitors tend to increase—the overall effect on threshold is indeterminate, pulled down-ward by the former and upward by the latter.

Model Validation

One can take the “standard” concave market share response model—that of the second (effective) regime—and augment it in a number of ways. To this standard (henceforth “base”) model, three con-structs have been grafted: (1) an “ineffective” regime;

(2) S-shaped response in advertising expenditure; and

(3) dynamic, multivariate regime switching _p_. We explore whether such additions are warranted by benchmarking the proposed model against an inter-related set of similar ones10:

M1: Single-Regime Concave (“Base”) Model.

Equivalent to (1) alone (or by (1) and (2) with p set to 0), i.e., assuming that there are no threshold effects, and thus only an effective regime.

M2: Single-Regime Parsimonious S-Shaped Model. Identical to the single-regime model, but with advertising entering (1) as exp −_A/_1 + Ait _ rather than A_it1 . This form is S-shaped only for _A > 2, thus allowing for S-shaped response—without overtly requiring it—with the same number of parameters as the single-regime model, so that no parsimony is sacrificed.

M3: Single-Regime Logistic Advertising Model.

Identical to the single-regime model, but with adver-tising entering (1) as a linear-logistic transform, _1 + exp_%0 +%1A__−1, rather than A_it1 . This form allows for a more general type of S-shaped advertising response than either of the other single-regime models, but is less parsimonious.12 The advertising specification is similar to that used by Mahajan and Muller (1987) and Feinberg (1992).

M4: Two-Regime Constant p Model (Bemmaor 1984). Equivalent to (1), but with p an estimated con-stant (that is, independent of exogenous variables).

10 As Category 3 lacks advertising thresholds, we limit model com-parisons to the two durables categories.

11 Because &2y/&A2 = _y_A_1 + Ait _−4___A − 2_1 + Ait __ and Ait is non-negative, there is an inflection point iff _A > 2.

12 Model M3 is equivalent to M7, under the restrictions

_c _ _ cN _ cE _ = 0 (M6), _A = 0, and _i = _i . To see that M3, M6, M7

_D _P _E _T and

and can capture S-shaped response, hold aside Dit Pit Ei Tit

suppress subscripts i and t:

Brand i’s market share at time t is

M ∝ 1 + exp_−_c1 + c____ _ 1 + exp_−_c0 + c____

where c1 = c0 − log_K1/K2_ and _ = log_A_. Simple algebra shows

&2M/&A2 = 0, when A = exp_−_log_c_ + 1_ − log_c_ − 1_ + c0_/c__. Thus, an inflection point occurs when c_ < −1, so S-shape response is possible, but not inevitable, for M3, M6, and M7.

M5: Two-Regime Cutoff Model. Equivalent to (1), but with the specification for p given by (4), i.e., a two-regime model in which thresholds are deterministic, and a single cutoff point is estimated (through a grid search) beneath which advertising is “ineffective.”

M6: Two-Regime p = f _A_ Model. Equivalent to

(1) and (2)—the full probabilistic thresholds model— but with the nonadvertising coefficients in (2), {c_ _ cN _ cE }, set to zero. The model therefore lacks category-specific and dynamic factors in the regime-switching probability (and is the model estimated for Category 3 in Table 1).

M7: Two-Regime Probabilistic Thresholds Model.

(1) and (2).

Although each of the six other models is related to the probabilistic thresholds model, only M4 (Bemmaor’s two-regime constant p model) and M6 (the two-regime p = f _A_ model) are strictly nested within it in the sense of parametric restriction.13 The remaining two-regime models, M4 and M5, are spe-cial cases of the probabilistic thresholds model for boundary values of p (respectively, always p = 0, or variously 0/1 depending on Ait , as per Equation (4)), and thus we use BIC to compare relative model per-formance. It is helpful to anticipate what insights might be gained through various comparisons among seven models:

Single- vs. two-regime models (M1–M3 vs. M4–M7)(Is a second regime warranted? S-shaped advertising response vs. concavity (M1 vs. M2 vs. M3)( Irrespective of threshold effects, is there evi-dence for S-shaped advertising response?

Dependence of the regime-switching probability (p) on advertising level (M4 vs. M5–M7)( Does the overall expenditure level help determine which of the two regimes is operational?

Specification of advertising response (M5 vs. M6–M7)(If advertising level appears in the regime-switching probability _p_ specification, what form should the functional relationship take?

Inclusion of dynamic marketing variables in specifica-tion for p (M7 vs. M4–M6)( Can a model that excludes time-varying and environmental factors capture the data adequately?

By structuring comparisons in this manner, a ratio-nale for a model such as the probabilistic thresholds will either be “built up” or fail to be, with each com-ponent of the model spotlighted as to its marginal importance. The results of all such comparisons can be found in Table 2.

13 It is possible to construe M1 (the single-regime concave model) in a similar manner by setting the four effects coefficients to zero and the constant term in Equation (2) to negative infinity. However, we apply the more conservative nonnested tests in this case.

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Table 2 Model Validation and Nested Model Comparison

Number of Parameters Log Likelihood BIC and (Rank)

Model Category 1 Category 2 Category 1 Category 2 Category 1 Category 2

M1: Single-regime concave 7 7 −2 996_1 −1 594_7 −3 026_4 _6_ −1 618_7 _7_

M2: Single-regime parsimonious S-shape 7 7 −3 024_1 −1 591_4 −3 054_4 _7_ −1 615_4 _6_

M3: Single-regime logistic advertising 8 8 −2 891_9 −1 586_3 −2 926_0 _4_ −1 613_7 _5_

M4: Two-regime constant p 13 13 −2 918_8 −1 409_1 −2 971_9 _5_ −1 453_7 _4_

M5: Two-regime cutoff (grid search) 14 14 −2 792_5 −1 316_3 −2 853_2 _3_ −1 364_4 _3_

M6: Two-regime p = f _A_ 14 14 −2 715_6 −1 276_0 −2 772_5 _2_ −1 324_1 _2_

M7: Probabilistic thresholds 17 17 −2 682_3 −1 245_1 −2 750_6 _1_ −1 303_5 _1_

Nested Model Comparisons _ df −2__LL_ p-Value for __2 df

M6 vs. M4 1 1 406_4 266_2 3.3 E-16 3.8 E-15

M7 vs. M4 4 4 473_0 328_0 1.2 E-13 1.2 E-12

M7 vs. M6 3 3 66_6 61_8 2.3 E-14 2.4 E-13

Model Validation Results

As can be seen from the “BIC (and Rank)” column of Table 2, the pattern of ranking results for the BIC measure is remarkably similar across the two durables categories,14 with the probabilistic thresh-olds model faring best (M7: BIC (within-category rank) = −2_750_6 _1_; −1_303_5 _1_), followed by the two-regime p = f _A_ model (M6: −2_772_5 _2_; −1_324_1 _2_), and with the single-regime concave model near last in both categories (M1: −3_026_4 _6_; −1_618_7 _ 7_). In order to provide structure to our discussion of model comparison, we follow the five questions put forth earlier. All comparisons are based on BIC (Table 2) unless otherwise noted (i.e., likeli-hood ratio test for nested models).

Single- vs. Two-Regime Models (M1–M3 vs. M4–M7). The two-regime models perform far better than their more parsimonious counterparts, particu-larly so in Category 2. This is not merely a matter of insufficiently penalizing nonparsimonious models; for example, in Category 1 the single-regime logistic advertising model (M3: −2_926_0 _4_) performs bet-ter than the two-regime constant p (M4: −2_971_9 _5_) model. However, this is but a single case, and all other comparisons provide ample evidence supporting the existence of an “ineffective” regime.

S-Shaped Advertising Response vs. Concavity (M1 vs. M2 vs. M3). To determine whether S-shaped response is warranted in the absence of a second regime, we compare the globally concave M1 to two variants for which advertising can be S-shaped. Within this set of three models, for both data sets a clear pattern emerges, with the globally concave model (M1: −3_026_4 _6_; −1_618_7 _7_) and the Both Kendall’s _ __ = 0_810, p < 0_01_ and Spearman’s ) _) = 0_929, p < 0_01_ reject a lack of rank-order association.

parsimonious S-shaped model (M2: −3_054_4 _7_; −1_615_4 _6_) performing least well, and the logistic advertising model (M3: −2_926_0 _4_; −1_613_7 _5_) besting each.

Dependence of the Regime-Switching Probabil-ity _p_ on Advertising Level (M4 vs. M5–M7).

Over and above any effects accounted for by the share models (1), regime-switching probabilities (2) are strongly affected by advertising-spending levels. For both categories, the BIC measure indicates a far poorer fit for the constant p model (M4: −2_971_9 _5_; −1_453_7 _4_) than for the other two-regime mod-els, the cutoff (M5: −2_853_2 _3_; −1_364_4 _3_), p = f _A_ (M6: −2_772_5 _2_; −1_324_1 _2_) and the prob-abilistic thresholds (M7: −2_750_6 _1_; −1_303_5 _1_) models (a simple Chi-square test on the nested mod-els M4 vs. M6 indicates the enormity of the differ-ence, with *2_1 df _ = 406_4 and 266.2, respectively, for Categories 1 and 2). This is especially compelling evidence that merely including advertising levels in the response functions _1_ alone fails to adequately account for market share dynamics in these cate-gories. Given the role played by c_ in the elasticity expression (Footnote 4), excluding advertising from the regime-switching probability expression can lead to biased elasticity measures.

Specification of Advertising Response (M5 vs. M6–M7): Having established that there should be two regimes and that switching probability depends on advertising levels, we wish to determine the nature of this functional dependence. Specifically, as ad spending diminishes, should the regimes switch abruptly or taper off? The cutoff model (M5) posits the former, seeking the point beneath which adver-tising ceases to be effective (given that this level depends on other variables as well). By contrast, the p = f _A_ (M6) and probabilistic thresholds (M7)

Vakratsas, Feinberg, Bass, and Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS

model entail continuous logistic relationship (2); in M6, advertising as the sole input variable. Again, there is a clear pattern, with both “continuous specifica-tion” models (M6: −2_772_5 _2_; −1_324_1 _2_; M7: −2_750_6 _1_; −1_303_5 _1_) edging out the “discrete-jump” cutoff model (M5: −2_853_2 _3_; −1_364_4 _3_).

Inclusion of Dynamic Marketing Variables in Specification for p (M7 vs. M4–M6). Among our main goals is to determine the existence of thresh-olds and whether they are influenced by time-variant, brand-specific factors. The only model of the seven capable of capturing this (over and above the other constructs previously addressed) is M7, the proba-bilistic thresholds model. Based again on BIC, it per-formed far better than all other models, including the other two-regime comparison models, which are only slightly more parsimonious. It (M7: −2_750_6 _1_; −1_303_5 _1_) offered performance well in excess of its nearest “competitor,” the p = f _A_ model (M6: −2_772_5 _2_; −1_324_1 _2_), a particularly telling com-parison in that M6 is a direct parametric restriction of M7, obtained by setting the coefficients for time in market, order of entry, and number of competitors to zero. Chi-squared tests—*2_3 df _ = 66_6 for Cate-gory 1 and 61.8 for Category 2—argue strongly for the inclusion of these omitted, dynamic factors.

This sequence of five comparisons, taken together, systematically supports our main line of argument: that threshold effects exist; that market share response to advertising is not necessarily globally concave; that which regime one operates in is a function of advertising level; and that dynamic, competitive, and brand-specific factors affect the regime-switching probability.

A number of additional insights emerge. First, it is intriguing to note that the cutoff model (M5) per-forms better than the constant p model (M4) for both categories. This may be attributed to the fact that, in both the probabilistic thresholds (M7) and the cut-off model (M5), a core construct—either the logistic probability (2) or the cutoff point (4), respectively— other than the market share model itself is guided by the level of advertising expenditures. For example, in the constant p model of Bemmaor (M4), although p is estimated for each data set, it does not vary with the level of advertising expenditure. While this mechanism does encode two different effectiveness regimes, it assumes that the likelihood of being in one or the other does not vary with expenditure; by con-trast, the cutoff model (M5) makes choice between the regimes completely a function of advertising level, in an all-or-nothing manner. In this way, the con-stant p and the cutoff model are rather like opposites, the former stating that regime-switching probability is not affected by advertising (or anything else), the latter saying that being above some critical advertis-ing threshold is literally all that matters. That neither performs as well as the proposed model may suggest that one should well allow for a gradual shift to the effective regime (e.g., as a function of market-level covariates, as in M7).

Consequently, the poor performance of the two-regime constant p model (in Category 1, where it barely bests the one-regime model) suggests not only the importance of a two-regime model, but of having the “switching function” (2) depend on strategic input variables such as advertising expenditure. A simi-lar comparison can be made between the proposed probabilistic thresholds model (M7) and the only other model which approached its performance, the two-regime cutoff model (M5). Whereas the cutoff model takes account of advertising, it does so in a rigid, abrupt fashion. The proposed model, by con-trast, allows for thresholds which are both dynamic and brand specific, assigning a probability of operat-ing in the effective regime for any set of mix vari-able inputs. Indeed, the degree of dominance (based on BIC) of the probabilistic thresholds model over the other two-regime models is decisive, and speaks to the additional explanatory power, specifically, of including dynamic competitive and marketing-mix variables in the logistic probability specification (4). Taken as a group, the degree of dominance of the three two-regime models suggests that the inclusion of an ineffective regime more than compensates for the lack of parsimony involved.

Of all the model comparisons, the one we believe to be most salient is that to the single-regime S-shaped model, for several reasons. First, as shown above (Footnote 12), the probabilistic thresholds model is capable of producing S-shaped response to adver-tising spending. Because each model allows for S-shaped response, their comparison boils down to whether including an ineffective regime is called for over and above S-shaped response, a major point of justi-fication for the present study. Second, because the S-shaped model requires the same number of param-eters as the single-regime (concave) model, and far fewer than the probabilistic thresholds model, its fit statistics give some sense of the relative additional explanatory power of S-shaped response vs. a second, ineffective regime. Based on BIC, the data suggest unequivocally, for both categories, that S-shaped response alone provides comparatively little in terms of fit, compared with a second, ineffective regime: Taking the single-regime concave model—no S-shape, no ineffective regime—as a baseline (M1: −3_026_4 _6_; −1_618_7 _7_), the differential in “adding” S-shaped response (M2: −3_054_4 _ 7_; −1_615_4 _6_) is far less than further adding a second regime, as in the probabilistic thresholds model (M7: −2_750_6 _1_; −1_303_5 _1_).

Vakratsas, Feinberg, Bass, and Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS

Parsimoniousness vs. Flexibility. The issue re-mains as to whether better fit of the two-regime model is attributable to threshold effects per se or just to greater “flexibility”—in the sense that the switching function allows the market share model to vary by firm, whereas the other models have a single function that is held fixed across firms—or its being less parsimonious. Thorough discussions of the relevant issues on model complexity can be found in Van Heerde et al. (2001) treatment of deal effects, as well as the “potentially nonmonotonic” spline approach to pricing of Kalyanam and Shively (1998). To rule out these possibilities, we estimate a less parsimonious S-shaped model, one which should provide better fit based solely on more parameters and at least equal, if not greater, flexibility. Such a model can be described compactly as M8 = pM1 + _1 − p_M3, that is, a logistically mediated combina-tion of two single-regime models—M1 (single-regime convex model), M3 (single-regime logistic model)— creating a hybrid more flexible, and less parsimo-nious, than M7 (two-regime probabilistic thresholds model). A simple way to think about this is as a model which allows S-shaped response directly in the “effective” regime, thus accommodating a wide range of response forms. Interestingly, the less parsimonious M8 fit worse in both durables categories.15 Specifi-cally, we find that, in terms of log likelihoods, for SUVs M7 = −2_682_3, M8 = −2_693_9; and for mini-vans M7 = −1_245_1, M8 = −1_251_0. Given this pat-tern of results for both categories, neither flexibility nor nonparsimoniousness alone accounts for the supe-rior fit of the proposed probabilistic thresholds model.

Sensitivity Analysis for the Switching Function.

One might question, if not the particular form of the switching probability, then the inclusion of three effects, order of entry (OE), time in market (TM), and number of competitors (NC), which may appear to intrinsically covary. To get a handle on this, we com-pared the “full” model (M7) to analogous models leaving out each of the variables, singly and in pairs.16 In the minivan category, leaving out any of the effects very significantly decreases fit (p < 0_0001), and for SUVs, each of the effects is at least marginally sig-nificant; moreover, most coefficients are stable across models. We stress that conclusions regarding multi-collinearity can be misleading, as it is important to

15 M8 requires 19 parameters; M7, 17. Because the models are nonnested—as they must be for the comparison to be a sensible test—this is entirely possible. Similar “convex combinations” of other models in the paper failed to improve results. Note as well that the comparison for the nondurable category (liquid detergent) is not relevant, as no advertising thresholds were detected in that case.

16 We thank the area editor for suggesting this analysis. Full results are available from the authors.

distinguish dynamic effects within brand and cross-sectional effects across them. It is possible that, say, NC can decrease overall while TM for a particular brand increases. So, although the (marginal) effect of each of the OE, NC, and TM on p is monotonic, combina-tions of the three can lead to a nonmonotonic dynamic threshold effect over time.

For the SUV category, results are clear cut: The dynamic effects are monotonically increasing, and TM and NC are both important. The situation for mini-vans, by contrast, is quite “rich;” the full (OE, NC, and TM) model does markedly better than all the other combinations because the net effect is nonmonotonic, and there is a trade-off between maturity (TM lowers threshold) and competition (NC increases threshold). We strongly believe that all three effects belong in a compelling model and, moreover, that potentially positively correlated variables like TM and NC being able to have opposing effects is an important and useful property.

Taken together, then, the pattern of results for these categories suggests that none of the constructs— S-shaped advertising response, additional parameters, a second regime, p as a function of advertising level— alone accounts for the far better fit of the probabilistic thresholds model, although each certainly exerts some effect. Rather, the dynamic, brand-specific nature of the thresholds themselves appears to strongly affect our ability to accurately model market share fluctua-tions in these categories. It is unfortunate that the data available do not allow formal cross-category com-parisons, particularly so in (nondurable) Category 3, where we find no evidence of advertising thresh-olds. We note, however, that cross-category data used for such purposes would need to be relatively free of category-specific idiosyncrasies, a daunting task both in terms of data requirements and modeling formulation.

Summary and Conclusions

That there may be spending “regions” that produce little or no advertising response is an issue of clear practical import. When a brand enters a market, par-ticularly a turbulent or developing one, it is typi-cally the case that some form of variation (cycling, blitzing, or pulsing) is used. However, due to the dynamic complexity of the competitive environment, it is difficult to tell whether the “low” parts of the pulse are better than not advertising at all. The verdict of the extant literature in marketing, economics, and operations research has been that no such expenditure regions exist. Simply put: In terms of resulting share, some advertising is always better than no advertising at all. Our results question that unequivocal verdict.

Vakratsas, Feinberg, Bass, and Kalyanaram: Advertising Response Functions Revisited Marketing Science 23(1), pp. 109–119, © 2004 INFORMS

We believe there to be at least two related reasons for prior studies’ not having found compelling evi-dence of threshold effects. The first is that posited throughout: that the product classes for which data were available were simply not suitable for the task. We would concur that threshold effects play a modest role for frequently purchased goods in mature mar-kets, and have therefore focused on categories in a state of flux, either new or characterized by entry and exit of major players. The second possible rea-son for nonconfirmation of thresholds has to do, we believe, with the specification of the so-called regime-switching probability. If the probability of falling into the “ineffective” region is fixed, or based solely on advertising level, one is essentially left with a single-regime model: All that really matters is advertising level, even if some formal probabilistic mechanism is involved. By making the regime-switching probability a function of dynamic and category-specific factors, threshold effects stand in comparatively bold relief, as evidenced by the consistently superior fit of the probabilistic thresholds model over restricted, static variants.

Unsurprisingly, spending more increases the likeli-hood of advertising “effectively” in the sense of sur-passing the (dynamic) threshold. Similarly, the longer a brand is in the market, the higher its probability of being above threshold (for a given level of advertis-ing expenditures), so that “older” brands have lower thresholds, all else equal. A larger number of com-petitors decreases the probability of surpassing the threshold, consistent with the argument regarding “clutter” and the difficulty of being heard above it. We find only modest evidence, in one of the three categories, that order of entry affects the likelihood of operating in the “effective” regime, and then in a direction contrary to expectations. We stress that each of these effects was confirmed—and only could have been confirmed—through a model supporting the exis-tence of an ineffective regime, one which is partially determined by dynamic, competitive, and environ-mental factors.

Because thresholds lack direct observability, we estimated a variety of functional forms and model types. Contingent on the best of these, we then looked at constrained submodels (to determine whether all the variables in the switching regression formulation are really necessary) and less parsimonious ones (to rule out flexibility as the main driver of the threshold result). Across these sets of comparisons, there was unequivocal support for the influence of dynamic, brand-specific marketing and environmental factors in determining critical advertising threshold levels.

While there was some degree of support for the additional explanatory power of S-shaped response (of two distinct varieties) in a single-regime modeling context, there was a far stronger indication in favor of adding a second, ineffective regime (which was shown able to capture S-shaped response on its own). There was also strong support for the influence of advertising both in the response and switching func-tions, suggesting that models which include advertis-ing spending as a regressor in a single-regime market share model may exhibit an a priori misspecification.

There are several issues which our study raises but has not attempted to address. For example, we have not considered advertising as an endogenous firm decision variable. A possible extension of the present modeling framework would offer a formal method to account for this type of endogeneity, extending the seminal investigations in this area of Bass (1969) and Bass and Parsons (1969), as well as the recent work of Villas-Boas and Winer (1999). Further, the plausibility of any explanations involving life-cycle stage should be more formally investigated through a cross-category analysis of advertising thresholds. Coupled with corresponding profit data, a simi-lar model could provide implementable, dynamic guidelines for allocating advertising funds.

Acknowledgments

The authors wish to express their gratitude to Doug Bow-man for his help with the data and extensive commentary.

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